ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 22 (2022) #P2.01 / 163–205
https://doi.org/10.26493/1855-3974.2352.eaf
(Also available at http://amc-journal.eu)
Tight relative t-designs on two shells in
hypercubes, and Hahn and Hermite
polynomials*
Eiichi Bannai
Faculty of Mathematics, Kyushu University (emeritus), Japan, and
National Center for Theoretical Sciences, National Taiwan University, Taiwan
Etsuko Bannai
National Center for Theoretical Sciences, National Taiwan University, Taiwan
Hajime Tanaka †
Research Center for Pure and Applied Mathematics, Graduate School of Information
Sciences, Tohoku University, Sendai 980-8579, Japan
Yan Zhu ‡
College of Science, University of Shanghai for Science and Technology,
Shanghai 200093, China
Received 3 June 2020, accepted 24 May 2021, published online 14 April 2022
Abstract
Relative t-designs in the n-dimensional hypercubeQn are equivalent to weighted regu-
lar t-wise balanced designs, which generalize combinatorial t-(n, k, λ) designs by allowing
multiple block sizes as well as weights. Partly motivated by the recent study on tight Eu-
clidean t-designs on two concentric spheres, in this paper we discuss tight relative t-designs
inQn supported on two shells. We show under a mild condition that such a relative t-design
induces the structure of a coherent configuration with two fibers. Moreover, from this struc-
ture we deduce that a polynomial from the family of the Hahn hypergeometric orthogonal
polynomials must have only integral simple zeros. The Terwilliger algebra is the main tool
to establish these results. By explicitly evaluating the behavior of the zeros of the Hahn
polynomials when they degenerate to the Hermite polynomials under an appropriate limit
*This work was also partially supported by the Research Institute for Mathematical Sciences at Kyoto Univer-
sity.
†Corresponding author. Hajime Tanaka was supported by JSPS KAKENHI Grant Numbers JP25400034 and
JP17K05156.
‡Yan Zhu was supported by NSFC Grant No. 11801353.
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
164 Ars Math. Contemp. 22 (2022) #P2.01 / 163–205
process, we prove a theorem which gives a partial evidence that the non-trivial tight relative
t-designs in Qn supported on two shells are rare for large t.
Keywords: Relative t-design, association scheme, coherent configuration, Terwilliger algebra, Hahn
polynomial, Hermite polynomial.
Math. Subj. Class. (2020): 05B30, 05E30, 33C45
1 Introduction
This paper is a contribution to the study of relative t-designs in Q-polynomial associa-
tion schemes. In the Delsarte theory [16], the concept of t-designs is introduced for arbi-
trary Q-polynomial association schemes. For the Johnson scheme J(n, k), the t-designs
in the sense of Delsarte are shown to be the same thing as the combinatorial t-(n, k, λ)
designs. There are similar interpretations of t-designs in some other important families of
Q-polynomial association schemes [16, 17, 19, 34, 41]. The concept of relative t-designs
is also due to Delsarte [18], and is a relaxation of that of t-designs. Relative t-designs can
again be interpreted in several cases, including J(n, k). For the n-dimensional hypercube
Qn (or the binary Hamming scheme H(n, 2)) which will be our central focus in this paper,
these are equivalent to the weighted regular t-wise balanced designs, which generalize the
combinatorial t-(n, k, λ) designs by allowing multiple block sizes as well as weights.
The Delsarte theory has a counterpart for the unit sphere Sn−1 in Rn, established
by Delsarte, Goethals, and Seidel [20]. The t-designs in Sn−1 are commonly called the
spherical t-designs, and are essentially the equally-weighted cubature formulas of degree
t for the spherical integration, a concept studied extensively in numerical analysis. Spher-
ical t-designs were later generalized to Euclidean t-designs by Neumaier and Seidel [35]
(cf. [21]). Euclidean t-designs are in general supported on multiple concentric spheres in
Rn, and it follows that we may think of them as the natural counterpart of relative t-designs
in Rn. This point of view was discussed in detail by Bannai and Bannai [3]. See also [7, 8].
The success and the depth of the theory of Euclidean t-designs (cf. [38]) has been one driv-
ing force for the recent research activity on relative t-designs in Q-polynomial association
schemes; see, e.g., [3, 5, 6, 7, 8, 9, 11, 32, 51, 53, 54].
A relative t-design in a Q-polynomial association scheme (X,R) is often defined as
a certain weighted subset of the vertex set X , i.e., a pair (Y, ω) of a subset Y of X and
a function ω : Y → (0,∞). We are given in advance a ‘base vertex’ x ∈ X , and (Y, ω)
gives a ‘degree-t approximation’ of the shells (or spheres or subconstituents) with respect
to x on which Y is supported. See Sections 2 and 3 for formal definitions. Bannai and
Bannai [3] proved a Fisher-type lower bound on |Y |, and we call (Y, ω) tight if it attains
this bound. We may remark that t must be even in this case. In this paper, we continue the
study (cf. [5, 9, 32, 51, 53]) of tight relative t-designs in the hypercubesQn, which are one
of the most important families of Q-polynomial association schemes. The Delsarte theory
directly applies to the tight relative t-designs in Qn supported on one shell, say, the kth
shell, as these are equivalent to the tight combinatorial t-(n, k, λ) designs. (We note that
the kth shell induces J(n, k).) Our aim is to extend this structure theory to those supported
on two shells. We may view the results of this paper roughly as counterparts to (part of) the
E-mail addresses: bannai@math.kyushu-u.ac.jp (Eiichi Bannai), et-ban@rc4.so-net.ne.jp (Etsuko Bannai),
htanaka@tohoku.ac.jp (Hajime Tanaka), zhuyan@usst.edu.cn (Yan Zhu)
E. Bannai et al.: Tight relative t-designs on two shells in hypercubes, . . . 165
results by Bannai and Bannai [2, 4] on tight Euclidean t-designs on two concentric spheres.
Let t = 2e be even. In Theorem 5.3, which is our first main result, we show under
a mild condition that a tight relative 2e-design in Qn supported on two shells induces the
structure of a coherent configuration with two fibers. Moreover, from this structure we de-
duce that a certain polynomial of degree e, known as a Hahn polynomial, must have only
integral simple zeros. We note that the case e = 1 was handled previously by Bannai,
Bannai, and Bannai [5]. The Hahn polynomials are a family of hypergeometric orthogonal
polynomials in the Askey scheme [31, Section 1.5], and that their zeros are integral pro-
vides quite a strong necessary condition on the existence of such relative 2e-designs. The
corresponding necessary condition for the tight combinatorial 2e-(n, k, λ) designs from
the Delsarte theory was used successfully by Bannai [1]; that is to say, he showed that, for
each given integer e ⩾ 5, there exist only finitely many non-trivial tight 2e-(n, k, λ) de-
signs, where n and k (and thus λ) vary. See also [22, 36, 52]. We extend Bannai’s method
to prove our second main result, Theorem 7.1, which presents a version of his theorem for
our case.
The sections other than Sections 5 and 7 are organized as follows. We collect the
necessary background material in Sections 2 and 3. Section 3 also includes a few general
results on relative t-designs in Q-polynomial association schemes. As in [6, 44], our main
tool in the analysis of relative t-designs is the Terwilliger algebra [46, 47, 48], which is
a non-commutative semisimple C-algebra containing the adjacency algebra. Section 4 is
devoted to detailed descriptions of the Terwilliger algebra ofQn. It is well known (cf. [30,
31]) that the Hahn polynomials (3F2) degenerate to the Hermite polynomials (2F0) by an
appropriate limit process, and a key in Bannai’s method above was to evaluate precisely the
behavior of the zeros of the Hahn polynomials in this process. In Section 6, we revisit this
part of the method in a form suited to our purpose. Our account will also be simpler than
that in [1]. In Appendix, we provide a proof of a number-theoretic result (Proposition 7.2)
which is a variation of a result of Schur [40, Satz I].
2 Coherent configurations and association schemes
We begin by recalling the concept of coherent configurations.
Definition 2.1. The pair (X,R) of a finite set X and a set R of non-empty subsets of X2
is called a coherent configuration on X if it satisfies the following (C1) – (C4):
(C1) R is a partition of X2.
(C2) There is a subset R0 of R such that⊔
R∈R0
R = {(x, x) : x ∈ X}.
(C3) R is invariant under the transposition τ : (x, y) 7→ (y, x) ((x, y) ∈ X2), i.e.,
Rτ ∈ R for all R ∈ R.
(C4) For all R,S, T ∈ R and (x, y) ∈ T , the number
pTR,S :=
∣∣{z ∈ X : (x, z) ∈ R, (z, y) ∈ S}∣∣
is independent of the choice of (x, y) ∈ T .
166 Ars Math. Contemp. 22 (2022) #P2.01 / 163–205
Moreover, a coherent configuration (X,R) on X is called homogeneous if |R0| = 1, and
an association scheme if Rτ = R for all R ∈ R.
Remark 2.2. Suppose that a finite group G acts on X , and let R be the set of the orbitals
of G, that is to say, the orbits of G in its natural action on X2. Then (X,R) is a coherent
configuration. Moreover, (X,R) is homogeneous (resp. an association scheme) if and only
if the action of G on X is transitive (resp. generously transitive, i.e., for any x, y ∈ X we
have (xg, yg) = (y, x) for some g ∈ G).
Let (X,R) be a coherent configuration as above. For every R ∈ R0, let ΦR be the
subset of X such that R = {(x, x) : x ∈ ΦR}. Then we have⊔
R∈R0
ΦR = X.
We call the ΦR (R ∈ R0) the fibers of (X,R). By setting in (C4) either R ∈ R0 and
S = T , or S ∈ R0 and R = T , it follows that for every T ∈ R, we have T ⊂ ΦR × ΦS
for some R,S ∈ R0. In particular, (X,R) is homogeneous whenever it is an association
scheme. Let
γR,S =
∣∣{T ∈ R : T ⊂ ΦR × ΦS}∣∣ (R,S ∈ R0).
The matrix
[γR,S ]R,S∈R0 ,
which is symmetric by (C3), is called the type of (X,R).
Let MX(C) be the C-algebra of all complex matrices with rows and columns indexed
by X , and let V = CX be the C-vector space of complex column vectors with coordinates
indexed by X . We endow V with the Hermitian inner product
⟨u, v⟩ = v†u (u, v ∈ V ),
where † denotes adjoint. For every R ∈ R, let AR ∈ MX(C) be the adjacency matrix of
the graph (X,R) (directed, in general), i.e.,
(AR)x,y =
{
1 if (x, y) ∈ R,
0 otherwise,
(x, y ∈ X).
Then (C1) – (C4) above are rephrased as follows:
(A1)
∑
R∈R
AR = J (the all-ones matrix).
(A2)
∑
R∈R0
AR = I (the identity matrix).
(A3) (AR)† ∈ {AS : S ∈ R} (R ∈ R).
(A4) ARAS =
∑
T∈R
pTR,SAT (R,S ∈ R).
E. Bannai et al.: Tight relative t-designs on two shells in hypercubes, . . . 167
Let
A = span{AR : R ∈ R}.
Then from (A2) and (A4) it follows that A is a subalgebra ofMX(C), called the adjacency
algebra of (X,R). We note that A is semisimple as it is closed under † by virtue of (A3).
By (A1), A is also closed under entrywise (or Hadamard or Schur) multiplication, which
we denote by ◦. The AR are the (central) primitive idempotents of A with respect to ◦, i.e.,
AR ◦AS = δR,SAR,
∑
R∈R
AR = J.
Remark 2.3. If (X,R) arises from a group action as in Remark 2.2, then A coincides with
the centralizer algebra (or Hecke algebra or commutant) for the corresponding permutation
representation g 7→ Pg (g ∈ G) on V , i.e.,
A = {B ∈MX(C) : BPg = PgB (g ∈ G)}.
A subalgebra of MX(C) is called a coherent algebra if it contains J , and is closed
under ◦ and †. We note that the coherent algebras are precisely the adjacency algebras of
coherent configurations. It is clear that the intersection of coherent algebras in MX(C) is
again a coherent algebra. In particular, for any subset S of MX(C), we can speak of the
smallest coherent algebra containing S, which we call the coherent closure of S.
From now on, we assume that (X,R) is an association scheme. As is the case for many
examples of association schemes, we write
R = {R0, R1, . . . , Rn}, where R0 = {R0},
and say that (X,R) has n classes. We will then abbreviate pki,j = p
Rk
Ri,Rj
, Ai = ARi ,
and so on. The adjacency algebra A is commutative in this case, and hence it has a basis
E0, E1, . . . , En consisting of the (central) primitive idempotents, i.e.,
EiEj = δi,jEi,
n∑
i=0
Ei = I.
Put differently, E0V,E1V, . . . , EnV are the maximal common eigenspaces (or homoge-
neous components or isotypic components) of A, and the Ei are the corresponding orthog-
onal projections. Since the Ai are real symmetric matrices, so are the Ei. Note that the
matrix |X|−1J ∈ A is an idempotent with rank one, and thus primitive. We will always
set
E0 =
1
|X|
J.
For convenience, we let
Ai = Ei := O (the zero matrix) if i < 0 or i > n.
Though our focus in this paper will be on Q-polynomial association schemes, we first
recall the P -polynomial property for completeness. We say that the association scheme
(X,R) is P -polynomial (or metric) with respect to the ordering A0, A1, . . . , An if there
168 Ars Math. Contemp. 22 (2022) #P2.01 / 163–205
are non-negative integers ai, bi, ci (0 ⩽ i ⩽ n) such that bn = c0 = 0, bi−1ci ̸= 0
(1 ⩽ i ⩽ n), and
A1Ai = bi−1Ai−1 + aiAi + ci+1Ai+1 (0 ⩽ i ⩽ n),
where b−1 and cn+1 are indeterminates. In this case, A1 recursively generates A, and
hence has n+ 1 distinct eigenvalues θ0, θ1, . . . , θn ∈ R, where we write
A1 =
n∑
i=0
θiEi. (2.1)
We note that (X,R) is P -polynomial as above precisely when the graph (X,R1) is a
distance-regular graph and (X,Ri) is the distance-i graph of (X,R1) (0 ⩽ i ⩽ n). See,
e.g., [10, 12, 27, 15] for more information on distance-regular graphs.
We say that (X,R) is Q-polynomial (or cometric) with respect to the ordering
E0, E1, . . . , En if there are real scalars a∗i , b
∗
i , c
∗
i (0 ⩽ i ⩽ n) such that b
∗
n = c
∗
0 = 0,
b∗i−1c
∗
i ̸= 0 (1 ⩽ i ⩽ n), and
E1 ◦ Ei =
1
|X|
(b∗i−1Ei−1 + a
∗
iEi + c
∗
i+1Ei+1) (0 ⩽ i ⩽ n), (2.2)
where b∗−1 and c
∗
n+1 are indeterminates. In this case, |X|E1 recursively generates A with
respect to ◦, and hence has n+ 1 distinct entries θ∗0 , θ∗1 , . . . , θ∗n ∈ R, where we write
|X|E1 =
n∑
i=0
θ∗iAi. (2.3)
We call the θ∗i the dual eigenvalues of |X|E1. We may remark that E1 ◦ Ei, being a
principal submatrix of E1 ⊗ Ei, is positive semidefinite, so that the scalars a∗i , b∗i , and c∗i
are non-negative (the so-called Krein condition). The Q-polynomial association schemes
are an important subject in their own right, and we refer the reader to [23, 29] and the
references therein for recent activity.
Below we give two fundamental examples ofP - andQ-polynomial association schemes,
both of which come from transitive group actions. See [10, 12, 16] for the details.
Example 2.4. Let v and k be positive integers with v > k, and letX be the set of k-subsets
of {1, 2, . . . , v}. Set n = min{k, v − k}. For x, y ∈ X and 0 ⩽ i ⩽ n, we let (x, y) ∈ Ri
if |x ∩ y| = k − i. The Ri are the orbitals of the symmetric group Sv acting on X . We
call (X,R) a Johnson scheme and denote it by J(v, k). The eigenvalues of A1 are given
in decreasing order by
θi = (k − i)(v − k − i)− i (0 ⩽ i ⩽ n),
and J(v, k) isQ-polynomial with respect to the corresponding ordering of theEi (cf. (2.1)).
Example 2.5. Let q ⩾ 2 be an integer and let X = {0, 1, . . . , q − 1}n. For x, y ∈ X and
0 ⩽ i ⩽ n, we let (x, y) ∈ Ri if x and y differ in exactly i coordinate positions. The Ri
are the orbitals of the wreath product Sq ≀Sn of the symmetric groups Sq and Sn acting
E. Bannai et al.: Tight relative t-designs on two shells in hypercubes, . . . 169
on X . We call (X,R) a Hamming scheme and denote it by H(n, q). The eigenvalues of
A1 are given in decreasing order by
θi = n(q − 1)− qi (0 ⩽ i ⩽ n),
andH(n, q) isQ-polynomial with respect to the corresponding ordering of theEi (cf. (2.1)).
The Hamming scheme H(n, 2) is also known as the n-cube (or n-dimensional hypercube)
and is denoted by Qn.
Assumption 2.6. For the rest of this section and in Section 3, we assume that (X,R) is
an association scheme and is Q-polynomial with respect to the ordering E0, E1, . . . , En of
the primitive idempotents.
In general, for any positive semidefinite Hermitian matrices B,C ∈ MX(C), we have
(cf. [45])
(B ◦ C)V = span(BV ◦ CV ),
where
BV ◦ CV = {u ◦ v : u ∈ BV, v ∈ CV }.
Hence it follows from (2.2) that
span(E1V ◦ EiV ) =
{
Ei−1V + EiV + Ei+1V if a∗i ̸= 0,
Ei−1V + Ei+1V if a∗i = 0,
(0 ⩽ i ⩽ n), (2.4)
from which it follows that
h∑
i=0
k∑
j=0
span(EiV ◦ EjV ) =
h∑
i=0
k∑
j=0
span(E1V ◦ · · · ◦ E1V︸ ︷︷ ︸
i times
◦EjV )
=
h+k∑
i=0
EiV (2.5)
for 0 ⩽ h, k ⩽ n. See also [10, Section 2.8].
We now fix a ‘base vertex’ x ∈ X . Let
Xi = {y ∈ X : (x, y) ∈ Ri} (0 ⩽ i ⩽ n).
We call the Xi the shells (or spheres or subconstituents) of (X,R) with respect to x. For
every i (0 ⩽ i ⩽ n), define the diagonal matrix E∗i = E
∗
i (x) ∈MX(C) by
(E∗i )y,y =
{
1 if y ∈ Xi,
0 otherwise,
(y ∈ X).
Then we have
E∗i E
∗
j = δi,jE
∗
i ,
n∑
i=0
E∗i = I.
We call the E∗i the dual idempotents of (X,R) with respect to x. The subspace
A∗ = A∗(x) = span{E∗0 , E∗1 , . . . , E∗n}
170 Ars Math. Contemp. 22 (2022) #P2.01 / 163–205
is then a subalgebra of MX(C), which we call the dual adjacency algebra of (X,R) with
respect to x. The Terwilliger algebra (or subconstituent algebra) of (X,R) with respect to
x is the subalgebra T = T (x) of MX(C) generated by A and A∗ [46, 47, 48]. We note
that T is semisimple as it is closed under †.
Remark 2.7. If (X,R) arises from a group action as in Remark 2.2, which we recall is
generously transitive in this case, then T is a subalgebra of the centralizer algebra for the
action of the stabilizer Gx of x in G. The two algebras are known to be equal, e.g., for
J(v, k) and H(n, q); see [25, 43].
For every subset Y of X , let Ŷ ∈ V be the characteristic vector of Y , i.e.,
(Ŷ )y =
{
1 if y ∈ Y,
0 otherwise,
(y ∈ X).
In particular, X̂ denotes the all-ones vector in V . We will simply write x̂ for the character-
istic vector of the singleton {x}. With this notation established, we have
X̂i = E
∗
i X̂ = Aix̂ (0 ⩽ i ⩽ n),
from which it follows that
T x̂ = span{X̂i : 0 ⩽ i ⩽ n} = span{Eix̂ : 0 ⩽ i ⩽ n}. (2.6)
The T -module T x̂ is easily seen to be irreducible with dimension n + 1 (cf. [46, Lem-
ma 3.6]), and is called the primary T -module.
We define the dual adjacency matrix A∗1 = A
∗
1(x) ∈MX(C) by (cf. (2.3))
A∗1 = |X|diagE1x̂ =
n∑
i=0
θ∗iE
∗
i . (2.7)
Since the θ∗i are mutually distinct, A
∗
1 generates A
∗. Moreover, since
A∗1v = |X|(E1x̂) ◦ v (v ∈ V ),
it follows from (2.4) that
EiA
∗
1Ej = O if |i− j| > 1 (0 ⩽ i, j ⩽ n). (2.8)
Let W be an irreducible T -module. We define the dual support W ∗s , the dual endpoint
r∗(W ), and the dual diameter d∗(W ) of W by
W ∗s = {i : EiW ̸= 0}, r∗(W ) = minW ∗s , d∗(W ) = |W ∗s | − 1,
respectively. We call W dual thin if dimEiW ⩽ 1 (0 ⩽ i ⩽ n). We note that the
primary T -module T x̂ is dual thin, and that it is a unique irreducible T -module up to
isomorphism which has dual endpoint zero or dual diameter n. The following lemma is an
easy consequence of (2.8):
Lemma 2.8 ([46, Lemma 3.12]). With reference to Assumption 2.6, write A∗1 = A∗1(x),
A∗ = A∗(x), T = T (x). Let W be an irreducible T -module and set r∗ = r∗(W ),
d∗ = d∗(W ). Then the following hold:
E. Bannai et al.: Tight relative t-designs on two shells in hypercubes, . . . 171
1. A∗1EiW ⊂ Ei−1W + EiW + Ei+1W (0 ⩽ i ⩽ n).
2. W ∗s = {r∗, r∗ + 1, . . . , r∗ + d∗}.
3. EiA∗1EjW ̸= 0 if |i− j| = 1 (r∗ ⩽ i, j ⩽ r∗ + d∗).
4. Suppose that W is dual thin. Then
i∑
h=0
Er∗+hW =
i∑
h=0
(A∗1)
hEr∗W (0 ⩽ i ⩽ d
∗).
In particular, W = A∗Er∗W .
3 Relative t-designs in Q-polynomial association schemes
In this section, we develop some general theory on relative t-designs in Q-polynomial
association schemes.
Recall Assumption 2.6. Throughout this section, we fix a base vertex x ∈ X , and write
E∗i = E
∗
i (x) (0 ⩽ i ⩽ n), A
∗
1 = A
∗
1(x), A
∗ = A∗(x), and T = T (x). In Introduction,
we meant by a weighted subset of X a pair (Y, ω) of a subset Y of X and a function
ω : Y → (0,∞). For convenience, however, we extend the domain of ω to X by setting
ω(y) = 0 for every y ∈ X\Y . We will also naturally identify V with the set of complex
functions on X , so that ω ∈ V and Y = suppω. In our discussions on relative t-designs,
we will often consider the set
L = LY = {ℓ : Y ∩Xℓ ̸= ∅}, (3.1)
and say that (Y, ω) is supported on
⊔
ℓ∈LXℓ.
For comparison, we begin with the algebraic definition of t-designs in (X,R) due to
Delsarte [16, 17].
Definition 3.1. A weighted subset (Y, ω) of X is called a t-design in (X,R) if Eiω = 0
for 1 ⩽ i ⩽ t.
Delsarte [18] generalized this concept as follows:
Definition 3.2. A weighted subset (Y, ω) of X is called a relative t-design in (X,R) (with
respect to x) if Eiω ∈ span{Eix̂} for 1 ⩽ i ⩽ t.
Remark 3.3. Delsarte introduced the concept of t-designs for subsets Y of X in [16], i.e.,
when ω = Ŷ , whereas in [17, 18] he mostly considered general (i.e., not necessarily non-
negative) non-zero vectors ω ∈ V in the discussions on t-designs and relative t-designs.
Some facts/results below, such as Examples 3.4 and 3.5, Proposition 3.6, and Theorem 3.8,
are still valid for general ω ∈ V , but the Fisher-type lower bound on |Y | = | suppω|
(cf. Theorem 3.9) makes sense only when ω is non-negative.
For the Johnson and Hamming schemes, Delsarte [16, 17, 18] showed that these alge-
braic concepts indeed have geometric interpretations:
Example 3.4. Let (X,R) be the Johnson scheme J(v, k) from Example 2.4. Then (Y, ω)
is a t-design if and only if, for every t-subset z of {1, 2, . . . , v}, the sum λz of the values
172 Ars Math. Contemp. 22 (2022) #P2.01 / 163–205
ω(y) over those y ∈ Y such that z ⊂ y, is a constant independent of z. On the other hand,
(Y, ω) is a relative t-design if and only if the above λz depends only on |x ∩ z|. We note
that (Y, Ŷ ) is a t-design if and only if Y is a t-(v, k, λ) design (cf. [13, Chapter II.4]) for
some λ.
Example 3.5. Let (X,R) be the Hamming scheme H(n, q) from Example 2.5. Then
(Y, ω) is a t-design if and only if, for every t-subset T of {1, 2, . . . , n} and every function
f : T → {0, 1, . . . , q − 1}, the sum λT ,f of the values ω(y) over those
y = (y1, y2, . . . , yn) ∈ Y such that yi = f(i) (i ∈ T ), is a constant independent of
the pair (T , f). On the other hand, (Y, ω) is a relative t-design if and only if the above
λT ,f depends only on |{i ∈ T : xi = f(i)}|, where x = (x1, x2, . . . , xn). We note that
(Y, Ŷ ) is a t-design if and only if the transpose of the |Y | × n matrix formed by arranging
the elements of Y (in any order) is an orthogonal array OA(|Y |, n, q, t) (cf. [13, Chapter
III.6]). For the case q = 2, i.e., for Qn, if we choose the base vertex as x = (0, 0, . . . , 0),
then (Y, Ŷ ) is a relative t-design if and only if Y is a regular t-wise balanced design of type
t-(n,L, λ) (cf. [38, Section 4.4]) for some λ, where L is from (3.1), and where we identify
the elements of X = {0, 1}n with their supports.
Similar results hold for some other important families of P - and Q-polynomial association
schemes; see, e.g., [17, 18, 19, 34, 41].
Proposition 3.6 (cf. [3, Theorem 4.5]). With reference to Assumption 2.6, let (Y, ω) be a
weighted subset supported on
⊔
ℓ∈LXℓ. Then we have
ω|T x̂ =
∑
ℓ∈L
⟨ω, X̂ℓ⟩
|Xℓ|
X̂ℓ, (3.2)
where ω|T x̂ denotes the orthogonal projection of ω on the primary T -module T x̂. More-
over, (Y, ω) is a relative t-design if and only if
⟨ω, v⟩ = ⟨ω|T x̂, v⟩ =
∑
ℓ∈L
⟨ω, X̂ℓ⟩
|Xℓ|
⟨X̂ℓ, v⟩
for every v ∈
∑t
i=0EiV .
Proof. Recall (2.6). The first part follows since the X̂i form an orthogonal basis of T x̂
with ∥X̂i∥2 = |Xi|. The second part is also immediate from
Eiω ∈ span{Eix̂} ⇐⇒ Eiω ∈ T x̂ ⇐⇒ Eiω|T x̂ = Eiω.
Remark 3.7. It is clear that (Xℓ, X̂ℓ) is a relative n-design for every 0 ⩽ ℓ ⩽ n. Hence,
if (Y, ω) is a relative t-design such that Xℓ ⊂ Y for some ℓ, and if ω is constant on Xℓ,
then the weighted subset (Y \Xℓ, (I − E∗ℓ )ω) obtained by discarding Xℓ from Y is again
a relative t-design. This observation is particularly important when applying Theorem 3.8
below; for example, we can always assume that 0 ̸∈ L.
The following is a slight generalization of Delsarte’s Assmus–Mattson theorem for Q-
polynomial association schemes [18, Theorem 8.4], and can also be viewed as a variation
of [9, Theorem 3.3], which in turn generalizes [28, Proposition 1]. See also [11]. The
proof is in fact identical to that of [44, Theorem 4.3], but we include it below because of
the potential importance of the result.
E. Bannai et al.: Tight relative t-designs on two shells in hypercubes, . . . 173
Theorem 3.8. With reference to Assumption 2.6, let (Y, ω) be a relative t-design supported
on
⊔
ℓ∈LXℓ. Then (Y ∩Xℓ, E∗ℓω) is a relative (t− |L|+ 1)-design for every ℓ ∈ L.
Proof. Let U = (T x̂)⊥ be the orthogonal complement of T x̂ in V , which we recall is the
sum of all the non-primary irreducible T -modules in V . On the one hand, we have
ω|U ∈
∑
ℓ∈L
E∗ℓU.
Since A∗1 generates A
∗ and has at most |L| distinct eigenvalues on this subspace (cf. (2.7)),
it follows that
A∗ω|U = span
{
ω|U , A∗1ω|U , . . . , (A∗1)|L|−1ω|U
}
. (3.3)
On the other hand, since E0U = 0, that (Y, ω) is a relative t-design is rephrased as
ω|U ∈
n∑
i=t+1
EiU.
Hence it follows from (2.8) and (3.3) that
A∗ω|U ⊂
|L|−1∑
k=0
(A∗1)
k
n∑
i=t+1
EiU ⊂
n∑
i=t−|L|+2
EiU.
In particular, for every ℓ ∈ L we have
E∗ℓω|U ∈
n∑
i=t−|L|+2
EiU.
In other words, (Y ∩Xℓ, E∗ℓω) is a relative (t− |L|+ 1)-design, as desired.
Bannai and Bannai [3, Theorem 4.8] established the following Fisher-type lower bound
on the size of a relative t-design with t even:
Theorem 3.9. With reference to Assumption 2.6, let (Y, ω) be a relative 2e-design (e ∈ N)
supported on
⊔
ℓ∈LXℓ. Then
|Y | ⩾ dim
(∑
ℓ∈L
E∗ℓ
)(
e∑
i=0
EiV
)
.
Definition 3.10. A relative 2e-design (Y, ω) is called tight if equality holds above.
Recall from Example 3.5 that the relative t-designs in the hypercubes are equivalent to
the weighted regular t-wise balanced designs.
Example 3.11. Let (X,R) be the n-cube Qn from Example 2.5. Xiang [51] showed that
if e ⩽ ℓ ⩽ n− e for every ℓ ∈ L, then
dim
(∑
ℓ∈L
E∗ℓ
)(
e∑
i=0
EiV
)
=
min{|L|−1,e}∑
i=0
(
n
e− i
)
. (3.4)
174 Ars Math. Contemp. 22 (2022) #P2.01 / 163–205
We may remark that (cf. [12, Theorem 9.2.1])
dimEiV =
(
n
i
)
(0 ⩽ i ⩽ n). (3.5)
See also [32] and [6, Theorem 2.7, Example 2.9].
Example 3.12. Consider a symmetric 2-(n + 1, k, λ) design (cf. [13, Chapter II.6]). Ob-
serve that removing a point yields a tight relative 2-design in Qn with L = {k − 1, k}.
Likewise, taking the complement of every block which contains a given point followed by
removing that point gives rise to a tight relative 2-design in Qn with L = {k, n+ 1− k}.
The complement of this is yet another example1 such that L = {k − 1, n − k}. See [32,
Section 3] and [50, Theorem 8]. Note that the weights are constant for these three exam-
ples. On the other hand, Bannai, Bannai, and Bannai [5, Theorem 2.2] showed that there
is a tight relative 2-design in Qn with L = {2, n/2} for n ≡ 6 (mod 8), provided that a
Hadamard matrix of order n/2 + 1 exists. This construction provides examples in which
the weights take two distinct values depending on the shells. See also [53].
Example 3.13. Working with the tight 4-(23, 7, 1) and 4-(23, 16, 52) designs instead of a
symmetric 2-(n+1, k, λ) design as in Example 3.12, we obtain four tight relative 4-designs
in Q22 with constant weight such that
L ∈
{
{6, 7}, {6, 15}, {7, 16}, {15, 16}
}
.
See [9, Theorem 6.3] and [32, Section 3].
Let (Y, ω) be a tight relative 2e-design supported on
⊔
ℓ∈LXℓ. Bannai, Bannai, and
Bannai [5, Theorem 2.1] showed that if the stabilizer of x in the automorphism group of
(X,R) acts transitively on each of the shells Xi then ω is constant on Y ∩ Xℓ for every
ℓ ∈ L. The next theorem generalizes this result by replacing group actions by combinatorial
regularity. Observe that the fibers of the coherent closure of T are in general finer than the
shells Xi.
Theorem 3.14. With reference to Assumption 2.6, let (Y, ω) be a tight relative 2e-design
(e ∈ N) supported on
⊔
ℓ∈LXℓ. For every ℓ ∈ L, the weight ω is constant on Y ∩ Xℓ
provided that Xℓ remains a fiber of the coherent closure of T .
Proof. Let (cf. (3.2))
D = diagω, D̃ = diagω|T x̂ =
∑
ℓ∈L
⟨ω, X̂ℓ⟩
|X̂ℓ|
E∗ℓ .
Note that D̃ ∈ T . Let F be the orthogonal projection onto BV , where
B =
√
D̃
e∑
i=0
Ei ∈ T .
Observe that
BV = (BB†)V,
1It seems that this construction is missing in [50, Theorem 8].
E. Bannai et al.: Tight relative t-designs on two shells in hypercubes, . . . 175
and that F is written as a polynomial in the Hermitian (in fact, real and symmetric) matrix
BB†. In particular, F ∈ T .
Since (Y, ω) is tight, we have
dimBV = dim
√
D̃
(∑
ℓ∈L
E∗ℓ
)(
e∑
i=0
EiV
)
= |Y |.
Let u1, u2, . . . , u|Y | be an orthonormal basis of BV , and let
G =
[
u1 u2 · · · u|Y |
]
.
Then we have
F = GG†. (3.6)
Let
D′ = D|Y×Y , D̃′ = D̃|Y×Y , F ′ = F |Y×Y , G′ = G|Y×{1,2,...,|Y |},
where |Y×Y etc. mean taking corresponding submatrices. Note that these are square matri-
ces, and that D′ and D̃′ are invertible. Then it follows that
(G′)†D′(D̃′)−1G′ = I|Y |. (3.7)
Indeed, since we may write
ui =
√
D̃ vi, where vi ∈
e∑
r=0
ErV (1 ⩽ i ⩽ |Y |),
it follows from (2.5) (applied to h = k = e) and Proposition 3.6 that the (i, j)-entry of the
LHS in (3.7) is equal to
(vi)
†Dvj = ⟨ω, vi ◦ vj⟩ = ⟨ω|T x̂, vi ◦ vj⟩ = (vi)†D̃vj = ⟨uj , ui⟩ = δi,j ,
where means complex conjugate. By (3.6) and (3.7), we have
I|Y | = D
′(D̃′)−1G′(G′)† = D′(D̃′)−1F ′,
so that
(D′)−1 = (D̃′)−1F ′. (3.8)
In particular, F ′ is a diagonal matrix.
Now, let ℓ ∈ L and suppose that Xℓ remains a fiber of the coherent closure of T . Then
the (y, y)-entry of F ∈ T is constant for y ∈ Xℓ (cf. (A1) and (A2)), and the same is true
for D̃. Hence it follows from (3.8) that ω(y) = Dy,y must be constant for y ∈ Y ∩ Xℓ.
This completes the proof.
176 Ars Math. Contemp. 22 (2022) #P2.01 / 163–205
4 The Terwilliger algebra of Qn
For the rest of this paper, we will focus on relative t-designs in the n-cube Qn from Ex-
ample 2.5. We will need detailed descriptions of the Terwilliger algebra ofQn and its irre-
ducible modules, and we collect these in this section. Thus, we assume that (X,R) = Qn,
where X = {0, 1}n. We again fix a base vertex x ∈ X , and write E∗i = E∗i (x)
(0 ⩽ i ⩽ n), A∗1 = A
∗
1(x), and T = T (x). The Q-polynomial ordering we consider
is the one given in Example 2.5.2
Proposition 4.1 (cf. [39, Section I.C]). We have
T = span{E∗i AjE∗k : 0 ⩽ i, j, k ⩽ n}. (4.1)
In particular, T is a coherent algebra.
Proof. The RHS in (4.1) is a subspace of T . Recall from Example 2.5 that Qn admits
the action of G = S2 ≀ Sn. The stabilizer Gx of x in G is isomorphic to Sn, and it is
immediate to see that every orbital of Gx is of the form
{(y, z) ∈ X ×X : (x, y) ∈ Ri, (y, z) ∈ Rj , (z, x) ∈ Rk}
for some i, j, and k, where the corresponding adjacency matrix is E∗i AjE
∗
k . Hence the
RHS in (4.1) agrees with the centralizer algebra for the action of Gx on X , which is a
coherent algebra; cf. Remark 2.3. Since T is generated by the Ai and the E∗i , the result
follows.
Lemma 4.2. For 0 ⩽ i, j, k ⩽ n, we have E∗i AjE∗k ̸= O if and only if
j ∈
{
|i− k|, |i− k|+ 2, |i− k|+ 4, . . . ,min{i+ k, 2n− i− k}
}
.
Proof. Routine.
Next we recall basic facts about the irreducible T -modules. Let W be an irreducible
T -module. We define the support Ws, the endpoint r(W ), and the diameter d(W ) of W
by
Ws = {i : E∗iW ̸= 0}, r(W ) = minWs, d(W ) = |Ws| − 1,
respectively. We call W thin if dimE∗iW ⩽ 1 (0 ⩽ i ⩽ n).
Theorem 4.3 (cf. [26]). Let W be an irreducible T -module and set r = r(W ),
r∗ = r∗(W ), d = d(W ), and d∗ = d∗(W ). Then W is thin, dual thin, and we have
r = r∗, d = d∗ = n− 2r, Ws =W ∗s = {r, r + 1, . . . , n− r}.
Moreover, the isomorphism class of W is determined by r.
Remark 4.4. Recall that the universal enveloping algebra U(sl2(C)) is defined by the
generators x, y, h and the relations
xy− yx = h, hx− xh = 2x, hy− yh = −2y.
2If n is even then Qn has another Q-polynomial ordering E0, En−1, E2, En−3, . . . in terms of the natural
ordering; cf. [10, p. 305].
E. Bannai et al.: Tight relative t-designs on two shells in hypercubes, . . . 177
There is a surjective homomorphism U(sl2(C))→ T such that (cf. [26, Lemma 7.5])
x 7→
n∑
i=1
Ei−1A
∗
1Ei, y 7→
n−1∑
i=0
Ei+1A
∗
1Ei, h 7→ A1.
Every irreducible T -module is then irreducible as an sl2(C)-module. We also obtain an-
other surjective homomorphism U(sl2(C)) → T by interchanging A1 and A∗1 and replac-
ing the Ei by the E∗i above; cf. [26, Lemma 5.3].
From now on, we fix an orthogonal irreducible decomposition
V =
⊕
W∈Λ
W (4.2)
of the standard module V . In view of Theorem 4.3, let
Λr = {W ∈ Λ : r(W ) = r∗(W ) = r} (0 ⩽ r ⩽ ⌊n/2⌋), (4.3)
and fix a unit vector vW ∈ ErW for each W ∈ Λr. Since
dimEiV =
∑
W∈Λ
dimEiW =
i∑
r=0
|Λr| (0 ⩽ i ⩽ ⌊n/2⌋) (4.4)
by Theorem 4.3, it follows from (3.5) that
|Λr| =
(
n
r
)
−
(
n
r − 1
)
(0 ⩽ r ⩽ ⌊n/2⌋).
It is known (cf. [26, Theorem 9.2]) that if W ∈ Λr then the vectors
E∗r vW , E
∗
r+1vW , . . . , E
∗
n−rvW (4.5)
form an orthogonal basis of W , called a standard basis of W . By [26, Lemma 6.6], we
also have
∥E∗i vW ∥2 =
(
n− 2r
i− r
)
∥E∗r vW ∥2 (r ⩽ i ⩽ n− r). (4.6)
We note that
1 = ∥vW ∥2 =
n−r∑
i=r
∥E∗i vW ∥2 = 2n−2r∥E∗r vW ∥2. (4.7)
For W,W ′ ∈ Λr, we observe that the linear map W →W ′ defined by
E∗i vW 7→ E∗i vW ′ (r ⩽ i ⩽ n− r)
is an isometric isomorphism of T -modules. Let
Ĕi,jr =
2n−2r√(
n−2r
i−r
)(
n−2r
j−r
) ∑
W∈Λr
(E∗i vW )(E
∗
j vW )
† (r ⩽ i, j ⩽ n− r). (4.8)
178 Ars Math. Contemp. 22 (2022) #P2.01 / 163–205
Then we have
(Ĕi,jr )
† = Ĕj,ir (r ⩽ i, j ⩽ n− r), (4.9)
and from (4.6) and (4.7) it follows that
Ĕi,jr Ĕ
i′,j′
r′ = δr,r′δj,i′Ĕ
i,j′
r
for 0 ⩽ r, r′ ⩽ ⌊n/2⌋, r ⩽ i, j ⩽ n − r, and r′ ⩽ i′, j′ ⩽ n − r′. By Theorem 4.3
and Wedderburn’s theorem (cf. [14, Section 3]), T is isomorphic to the direct sum of full
matrix algebras
T ∼=
⌊n/2⌋⊕
r=0
Mn−2r+1(C),
and the Ĕi,jr form an orthogonal basis of T . See also [24, Section 2]. We note that
E∗i TE
∗
j = span
{
Ĕi,jr : 0 ⩽ r ⩽ min{i, j, n− i, n− j}
}
(0 ⩽ i, j ⩽ n). (4.10)
We now recall the Hahn polynomials [31, Section 1.5]
Qr(ξ;α, β,N) = 3F2
(
−ξ,−r, r + α+ β + 1
α+ 1,−N
∣∣∣∣ 1) ∈ R[ξ] (0 ⩽ r ⩽ N), (4.11)
where
sFt
(
a1, . . . , as
b1, . . . , bt
∣∣∣∣ c) = ∞∑
i=0
(a1)i · · · (as)i
(b1)i · · · (bt)i
ci
i!
,
and
(a)i = a(a+ 1) · · · (a+ i− 1).
For α, β > −1, or for α, β < −N , we have
N∑
ξ=0
(
α+ ξ
ξ
)(
β +N − ξ
N − ξ
)
Qr(ξ;α, β,N)Qr′(ξ;α, β,N)
= δr,r′
(−1)r(r + α+ β + 1)N+1(β + 1)rr!
(2r + α+ β + 1)(α+ 1)r(−N)rN !
. (4.12)
Our aim is to describe the entries of the Ĕi,jr . In view of (4.9), we will assume for the
rest of this section that
0 ⩽ i ⩽ j ⩽ n.
By Proposition 4.1 and Lemma 4.2, we have
E∗i TE
∗
j = span
{
E∗i A2ξ+j−iE
∗
j : 0 ⩽ ξ ⩽ min{i, n− j}
}
.
Moreover, it follows that (cf. (4.10))
E∗i A2ξ+j−iE
∗
j
=
min{i,n−j}∑
r=0
3F2
(
−ξ,−r, r − n− 1
j − n,−i
∣∣∣∣ 1)
(
j
i−ξ
)(
n−j
ξ
)(
j−r
j−i
)√(
n−2r
j−r
)
(
j
i
)√(
n−2r
i−r
) Ĕi,jr . (4.13)
E. Bannai et al.: Tight relative t-designs on two shells in hypercubes, . . . 179
This formula can be found in [33, Section 10]. See also [39, 49] for similar calculations.
If i ⩽ n− j then
3F2
(
−ξ,−r, r − n− 1
j − n,−i
∣∣∣∣ 1) = Qr(ξ; j − n− 1,−j − 1, i).
Since (
j
i− ξ
)(
n− j
ξ
)
= (−1)i
(
j − n− 1 + ξ
ξ
)(
−j − 1 + i− ξ
i− ξ
)
,
it follows from (4.12) (applied to α = j − n− 1, β = −j − 1, N = i) and (4.13) that, for
0 ⩽ r ⩽ i,
i∑
ξ=0
3F2
(
−ξ,−r, r − n− 1
j − n,−i
∣∣∣∣ 1)E∗i A2ξ+j−iE∗j
=
(−1)r(r − n− 1)i+1(−j)rr!
(2r − n− 1)(j − n)r(−i)ri!
·
(−1)i
(
j−r
j−i
)√(
n−2r
j−r
)
(
j
i
)√(
n−2r
i−r
) Ĕi,jr
=
(
n
i
)(
n−i
r
)√(
n−2r
j−r
)((
n
r
)
−
(
n
r−1
))(
n−j
r
)√(
n−2r
i−r
) Ĕi,jr .
Likewise, if n− j ⩽ i then
3F2
(
−ξ,−r, r − n− 1
j − n,−i
∣∣∣∣ 1) = Qr(ξ;−i− 1, i− n− 1, n− j).
In this case, since(
j
i− ξ
)(
n− j
ξ
)(
j
i
)−1
= (−1)n−j
(
−i− 1 + ξ
ξ
)(
i− 1− j − ξ
n− j − ξ
)(
n− i
n− j
)−1
,
again it follows from (4.12) (applied to α = −i− 1, β = i−n− 1, N = n− j) and (4.13)
that, for 0 ⩽ r ⩽ n− j,
n−j∑
ξ=0
3F2
(
−ξ,−r, r − n− 1
j − n,−i
∣∣∣∣ 1)E∗i A2ξ+j−iE∗j
=
(−1)r(r − n− 1)n−j+1(i− n)rr!
(2r − n− 1)(−i)r(j − n)r(n− j)!
·
(−1)n−j
(
j−r
j−i
)√(
n−2r
j−r
)
(
n−i
n−j
)√(
n−2r
i−r
) Ĕi,jr
=
(
n
i
)(
n−i
r
)√(
n−2r
j−r
)((
n
r
)
−
(
n
r−1
))(
n−j
r
)√(
n−2r
i−r
) Ĕi,jr .
In either case, it follows that
Ĕi,jr =
((
n
r
)
−
(
n
r−1
))(
n−j
r
)√(
n−2r
i−r
)
(
n
i
)(
n−i
r
)√(
n−2r
j−r
)
180 Ars Math. Contemp. 22 (2022) #P2.01 / 163–205
×
min{i,n−j}∑
ξ=0
3F2
(
−ξ,−r, r − n− 1
j − n,−i
∣∣∣∣ 1)E∗i A2ξ+j−iE∗j (4.14)
for 0 ⩽ i ⩽ j ⩽ n and 0 ⩽ r ⩽ min{i, n− j}.
5 Tight relative 2e-designs on two shells in Qn
We retain the notation of the previous sections. In this section, we discuss tight relative
2e-designs (Y, ω) in Qn supported on two shells Xℓ ⊔ Xm, i.e., L = {ℓ,m} (cf. (3.1)).
Recall from (3.4) that we have in this case
|Y | =
(
n
e
)
+
(
n
e− 1
)
,
but recall also that this is valid under the additional condition that e ⩽ ℓ,m ⩽ n− e. How-
ever, both (Y ∩Xℓ, E∗ℓω) and (Y ∩Xm, E∗mω) are relative (2e− 1)-designs by Theorem
3.8, so that if ℓ < 2e or ℓ > n − 2e for example, then (Y ∩ Xℓ, E∗ℓω) must be trivial in
view of Example 3.5, i.e., Xℓ ⊂ Y and ω is constant on Xℓ, and hence (Y ∩Xm, E∗mω) is
by itself a relative 2e-design; cf. Remark 3.7. This shows that the above condition is not a
restrictive one. We also note that
Lemma 5.1. Let (Y, ω) be a relative t-design inQn supported on
⊔
ℓ∈LXℓ. Then (Y
′, Anω)
is a relative t-design supported on
⊔
ℓ∈LXn−ℓ, where Y
′ = {y′ : y ∈ Y }, and for every
y ∈ X , y′ denotes the unique vertex such that (y, y′) ∈ Rn.
Proof. Immediate from EiAn ∈ span{Ei} (0 ⩽ i ⩽ n).
In view of the above comments, we now make the following assumption:
Assumption 5.2. In this section, let (Y, ω) be a tight relative 2e-design (e ∈ N) in Qn
supported on two shells Xℓ ⊔Xm, where
e ⩽ ℓ < m ⩽ n− ℓ (⩽ n− e).
Our aim is to show that Y then induces the structure of a coherent configuration with
two fibers, and to obtain a necessary condition on the existence of such (Y, ω) akin to
Delsarte’s theorem on tight 2e-designs. To this end, we first recall the proof of (3.4) given
in [6, Theorem 2.7, Example 2.9] under the above assumption.
For convenience, set
E∗L = E
∗
ℓ + E
∗
m.
By (4.2) and (4.3), we have
E∗L
(
e∑
i=0
EiV
)
=
e∑
r=0
∑
W∈Λr
E∗L
(
e∑
i=r
EiW
)
. (5.1)
Let W ∈ Λr, where 0 ⩽ r ⩽ e. Recall Theorem 4.3 and also the standard basis (4.5) of
W . If r = e then EeW is spanned by vW , and hence we have
E∗LEeW = span{E∗LvW }.
E. Bannai et al.: Tight relative t-designs on two shells in hypercubes, . . . 181
Note that E∗LvW is non-zero by Assumption 5.2, and hence
dimE∗LEeW = 1
in this case. Suppose next that 0 ⩽ r < e. On the one hand, since
E∗L
(
e∑
i=r
EiW
)
⊂ E∗LW = E∗ℓW + E∗mW,
we have
dimE∗L
(
e∑
i=r
EiW
)
⩽ 2.
On the other hand, it follows from (2.8) that
vW , A
∗
1vW ∈ ErW + Er+1W ⊂
e∑
i=r
EiW, (5.2)
and hence
E∗LvW , E
∗
LA
∗
1vW ∈ E∗L
(
e∑
i=r
EiW
)
.
Moreover, we have (cf. (2.7))
E∗LvW = E
∗
ℓ vW + E
∗
mvW , E
∗
LA
∗
1vW = θ
∗
ℓE
∗
ℓ vW + θ
∗
mE
∗
mvW ,
so that these two vectors are non-zero and are linearly independent by Assumption 5.2 and
since θ∗ℓ ̸= θ∗m. It follows that
dimE∗L
(
e∑
i=r
EiW
)
= 2.
Note that in this case we in fact have
E∗L
(
e∑
i=r
EiW
)
= span{E∗ℓ vW , E∗mvW },
as
E∗ℓ vW = E
∗
L
θ∗mI −A∗1
θ∗m − θ∗ℓ
vW , E
∗
mvW = E
∗
L
θ∗ℓ I −A∗1
θ∗ℓ − θ∗m
vW . (5.3)
Combining these comments, we now obtain (3.4) as follows:
dimE∗L
(
e∑
i=0
EiV
)
=
e∑
r=0
∑
W∈Λr
dimE∗L
(
e∑
i=r
EiW
)
= |Λe|+
e−1∑
r=0
2|Λr|
= dimEeV + dimEe−1V
182 Ars Math. Contemp. 22 (2022) #P2.01 / 163–205
=
(
n
e
)
+
(
n
e− 1
)
,
where we have used (3.5) and (4.4).
By the above discussions, the set of vectors below forms an orthogonal basis of the
subspace (5.1): (
e−1⊔
r=0
⊔
W∈Λr
{E∗ℓ vW , E∗mvW }
)⊔( ⊔
W∈Λe
{E∗LvW }
)
.
As in the proof of Theorem 3.14, let
D = diagω.
We next apply
√
D to the above basis vectors and compute their inner products. First, let
W,W ′ ∈
⊔e−1
r=0 Λr. It is clear that〈√
DE∗ℓ vW ,
√
DE∗mvW ′
〉
=
〈√
DE∗mvW ,
√
DE∗ℓ vW ′
〉
= 0. (5.4)
By (5.3), we have
(E∗ℓ vW ) ◦ (E∗ℓ vW ′) = E∗Lu,
where means complex conjugate, and
u =
(
θ∗mI −A∗1
θ∗m − θ∗ℓ
vW
)
◦
(
θ∗mI −A∗1
θ∗m − θ∗ℓ
vW ′
)
.
Observe that u belongs to
∑2e
i=0EiV by (2.5) (applied to h = k = e) and (5.2). Hence, by
Proposition 3.6 we have〈√
DE∗ℓ vW ,
√
DE∗ℓ vW ′
〉
= ⟨ω,E∗Lu⟩
= ⟨ω, u⟩
=
⟨ω, X̂ℓ⟩
|Xℓ|
⟨X̂ℓ, u⟩+
⟨ω, X̂m⟩
|Xm|
⟨X̂m, u⟩
=
⟨ω, X̂ℓ⟩
|Xℓ|
⟨X̂ℓ, E∗Lu⟩
=
⟨ω, X̂ℓ⟩
|Xℓ|
⟨E∗ℓ vW , E∗ℓ vW ′⟩
= δW,W ′
⟨ω, X̂ℓ⟩
|Xℓ|
∥E∗ℓ vW ∥2. (5.5)
Likewise, we have〈√
DE∗mvW ,
√
DE∗mvW ′
〉
= δW,W ′
⟨ω, X̂m⟩
|Xm|
∥E∗mvW ∥2. (5.6)
Next, let W ∈
⊔e−1
r=0 Λr and W
′ ∈ Λe. Then, by the same argument we have〈√
DE∗ℓ vW ,
√
DE∗LvW ′
〉
=
〈√
DE∗mvW ,
√
DE∗LvW ′
〉
= 0. (5.7)
E. Bannai et al.: Tight relative t-designs on two shells in hypercubes, . . . 183
Finally, let W,W ′ ∈ Λe. In this case, we have〈√
DE∗LvW ,
√
DE∗LvW ′
〉
= δW,W ′
(
⟨ω, X̂ℓ⟩
|X̂ℓ|
∥E∗ℓ vW ∥2 +
⟨ω, X̂m⟩
|X̂m|
∥E∗mvW ∥2
)
. (5.8)
Since (Y, ω) is a tight relative 2e-design, it follows from (5.4) – (5.8) that the set of vectors
below is an orthogonal basis of the subspace
√
DV = span{ŷ : y ∈ Y } of dimension
|Y | =
(
n
e
)
+
(
n
e−1
)
:(
e−1⊔
r=0
⊔
W∈Λr
{√
DE∗ℓ vW ,
√
DE∗mvW
})⊔( ⊔
W∈Λe
{√
DE∗LvW
})
.
For convenience, set
Yℓ = Y ∩Xℓ, Ym = Y ∩Xm.
We will naturally make the following identification by discarding irrelevant entries:
√
DE∗ℓ V = span{ŷ : y ∈ Yℓ} ←→ CYℓ ,√
DE∗mV = span{ŷ : y ∈ Ym} ←→ CYm .
We write
Λr =
{
W 1r ,W
2
r , . . . ,W
|Λr|
r
}
(0 ⩽ r ⩽ e).
For 0 ⩽ r ⩽ e, define a |Yℓ| × |Λr| matrix Hℓr and a |Ym| × |Λr| matrix Hmr by
Hℓr =
[√
DE∗ℓ vW 1r · · ·
√
DE∗ℓ vW |Λr|r
]
,
Hmr =
[√
DE∗mvW 1r · · ·
√
DE∗mvW |Λr|r
]
.
We then define a characteristic matrix H of (Y, ω) by
H =
[
Hℓ0 · · · Hℓe−1 O · · · O Hℓe
O · · · O Hm0 · · · Hme−1 Hme
]
.
We note that H is a square matrix of size |Y | =
(
n
e
)
+
(
n
e−1
)
. By (4.6), (4.7), and
(5.4) – (5.8), and since
|Xi| =
(
n
i
)
(0 ⩽ i ⩽ n),
we have
H†H =
(
e−1
⊕
r=0
κℓrI|Λr|
)
⊕
(
e−1
⊕
r=0
κmr I|Λr|
)
⊕ κeI|Λe|, (5.9)
where
κℓr =
ωℓ
(
n−2r
ℓ−r
)
2n−2r
(
n
ℓ
) , κmr = ωm
(
n−2r
m−r
)
2n−2r
(
n
m
) (0 ⩽ r < e),
184 Ars Math. Contemp. 22 (2022) #P2.01 / 163–205
κe =
ωℓ
(
n−2e
ℓ−e
)
2n−2e
(
n
ℓ
) + ωm(n−2em−e)
2n−2e
(
n
m
) ,
and we abbreviate
ωℓ = ⟨ω, X̂ℓ⟩, ωm = ⟨ω, X̂m⟩.
Let K denote the diagonal matrix on the RHS in (5.9). Then it follows that
I|Y | = HK
−1H†
=
[ ∑e
r=0
1
κℓr
Hℓr(H
ℓ
r)
† 1
κe
Hℓe(H
m
e )
†
1
κe
Hme (H
ℓ
e)
† ∑e
r=0
1
κmr
Hmr (H
m
r )
†
]
, (5.10)
where we write
κℓe = κ
m
e := κe (5.11)
for brevity. In particular, we have
1
κe
Hℓe(H
m
e )
† = O. (5.12)
Moreover, from (5.9) and (5.10) it follows that(
1
κℓr
Hℓr(H
ℓ
r)
†
)(
1
κℓr′
Hℓr′(H
ℓ
r′)
†
)
= δr,r′
1
κℓr
Hℓr(H
ℓ
r)
† (0 ⩽ r, r′ < e), (5.13)
1
κe
Hℓe(H
ℓ
e)
† = I|Yℓ| −
e−1∑
r=0
1
κℓr
Hℓr(H
ℓ
r)
†, (5.14)(
1
κmr
Hmr (H
m
r )
†
)(
1
κmr′
Hmr′ (H
m
r′ )
†
)
= δr,r′
1
κmr
Hmr (H
m
r )
† (0 ⩽ r, r′ < e), (5.15)
1
κe
Hme (H
m
e )
† = I|Ym| −
e−1∑
r=0
1
κmr
Hmr (H
m
r )
†. (5.16)
Note that the matrices (κℓr)
−1Hℓr(H
ℓ
r)
†, (κmr )
−1Hmr (H
m
r )
† (0 ⩽ r < e) are non-zero since
Hℓr , H
m
r are non-zero. Likewise, by setting
κr =
√
κℓrκ
m
r (0 ⩽ r < e)
for brevity, we have(
1
κℓr
Hℓr(H
ℓ
r)
†
)(
1
κr′
Hℓr′(H
m
r′ )
†
)
= δr,r′
1
κr
Hℓr(H
m
r )
† (0 ⩽ r, r′ < e), (5.17)(
1
κr
Hℓr(H
m
r )
†
)(
1
κmr′
Hmr′ (H
m
r′ )
†
)
= δr,r′
1
κr
Hℓr(H
m
r )
† (0 ⩽ r, r′ < e), (5.18)(
1
κr
Hℓr(H
m
r )
†
)(
1
κr′
Hmr′ (H
ℓ
r′)
†
)
= δr,r′
1
κℓr
Hℓr(H
ℓ
r)
† (0 ⩽ r, r′ < e), (5.19)(
1
κr
Hmr (H
ℓ
r)
†
)(
1
κr′
Hℓr′(H
m
r′ )
†
)
= δr,r′
1
κmr
Hmr (H
m
r )
† (0 ⩽ r, r′ < e). (5.20)
E. Bannai et al.: Tight relative t-designs on two shells in hypercubes, . . . 185
Since the matrices (κℓr)
−1Hℓr(H
ℓ
r)
†, (κmr )
−1Hmr (H
m
r )
† (0 ⩽ r < e) are non-zero, it fol-
lows from (5.17) – (5.20) that the matrices (κr)−1Hℓr(H
m
r )
† (0 ⩽ r < e) are non-zero and
are linearly independent.
It follows from Theorem 3.14 and Proposition 4.1 that ω is constant on each of Yℓ and
Ym, from which it follows that
Dy,y = ω(y) =

ωℓ
|Yℓ|
if y ∈ Yℓ,
ωm
|Ym|
if y ∈ Ym.
(5.21)
Hence, by comparing with the formula (4.8) for the matrices Ĕi,jr , we have
1
κℓr
Hℓr(H
ℓ
r)
† =
(
n
ℓ
)
|Yℓ|
Ĕℓ,ℓr |Yℓ×Yℓ (0 ⩽ r < e), (5.22)
1
κe
Hℓe(H
ℓ
e)
† =
ωℓ
(
n−2e
ℓ−e
)
2n−2eκe|Yℓ|
Ĕℓ,ℓe |Yℓ×Yℓ , (5.23)
1
κmr
Hmr (H
m
r )
† =
(
n
m
)
|Ym|
Ĕm,mr |Ym×Ym (0 ⩽ r < e), (5.24)
1
κe
Hme (H
m
e )
† =
ωm
(
n−2e
m−e
)
2n−2eκe|Ym|
Ĕm,me |Ym×Ym , (5.25)
1
κr
Hℓr(H
m
r )
† =
√(
n
ℓ
)(
n
m
)√
|Yℓ||Ym|
Ĕℓ,mr |Yℓ×Ym (0 ⩽ r < e), (5.26)
1
κe
Hℓe(H
m
e )
† =
√
ωℓωm
(
n−2e
ℓ−e
)(
n−2e
m−e
)
2n−2eκe
√
|Yℓ||Ym|
Ĕℓ,me |Yℓ×Ym , (5.27)
where |Yℓ×Yℓ etc. mean taking corresponding submatrices. From (5.23) and (5.25) it fol-
lows that the matrices (κe)−1Hℓe(H
ℓ
e)
†, (κe)
−1Hme (H
m
e )
† are also non-zero, since each of
Ĕℓ,ℓe |Yℓ×Yℓ , Ĕm,me |Ym×Ym has non-zero constant diagonal entries by (4.14).
Let H ′ be the set consisting of the |Y | × |Y | matrices of the form[ ∑e
r=0 a
ℓ,ℓ
r
1
κℓr
Hℓr(H
ℓ
r)
† ∑e−1
r=0 a
ℓ,m
r
1
κr
Hℓr(H
m
r )
†∑e−1
r=0 a
m,ℓ
r
1
κr
Hmr (H
ℓ
r)
† ∑e
r=0 a
m,m
r
1
κmr
Hmr (H
m
r )
†
]
,
where aℓ,ℓr etc. are in C, and we are again using the notation (5.11). By (5.13) – (5.20) and
the above comments, H ′ is a C-algebra with
dimH ′ = 4e+ 2. (5.28)
Define
Sℓ,ℓ(Y ) =
{
j : Rj ∩ (Yℓ × Yℓ) ̸= ∅
}
,
and define Sℓ,m(Y )(= Sm,ℓ(Y )) and Sm,m(Y ) in the same manner. Let H be the set
consisting of the |Y | × |Y | matrices of the form[ ∑
j∈Sℓ,ℓ(Y ) b
ℓ,ℓ
j Aj |Yℓ×Yℓ
∑
j∈Sℓ,m(Y ) b
ℓ,m
j Aj |Yℓ×Ym∑
j∈Sm,ℓ(Y ) b
m,ℓ
j Aj |Ym×Yℓ
∑
j∈Sm,m(Y ) b
m,m
j Aj |Ym×Ym
]
, (5.29)
186 Ars Math. Contemp. 22 (2022) #P2.01 / 163–205
where bℓ,ℓj etc. are in C. Then H is a C-vector space with
dimH = |Sℓ,ℓ(Y )|+ |Sℓ,m(Y )|+ |Sm,ℓ(Y )|+ |Sm,m(Y )|. (5.30)
Note that H is closed under ◦. By (5.22) – (5.26) and Proposition 4.1 (or (4.14)), H ′ is a
subspace of H .
By (4.14), (5.14), (5.22), and (5.23), we have
I|Yℓ| =
e−1∑
r=0
(
n
ℓ
)
|Yℓ|
Ĕℓ,ℓr |Yℓ×Yℓ +
ωℓ
(
n−2e
ℓ−e
)
2n−2eκe|Yℓ|
Ĕℓ,ℓe |Yℓ×Yℓ
=
1
|Yℓ|
min{ℓ,n−ℓ}∑
ξ=0
 e−1∑
r=0
((
n
r
)
−
(
n
r − 1
))
3F2
(
−ξ,−r, r − n− 1
ℓ− n,−ℓ
∣∣∣∣ 1)
+
ωℓ
(
n−2e
ℓ−e
)((
n
e
)
−
(
n
e−1
))
2n−2eκe
(
n
ℓ
) 3F2(−ξ,−e, e− n− 1ℓ− n,−ℓ
∣∣∣∣ 1)
A2ξ|Yℓ×Yℓ . (5.31)
Hence it follows that {ξ ̸= 0 : 2ξ ∈ Sℓ,ℓ(Y )} is a set of zeros of the polynomial
ψℓ,ℓe (ξ) =
e−1∑
r=0
((
n
r
)
−
(
n
r − 1
))
3F2
(
−ξ,−r, r − n− 1
ℓ− n,−ℓ
∣∣∣∣ 1)
+
ωℓ
(
n−2e
ℓ−e
)((
n
e
)
−
(
n
e−1
))
2n−2eκe
(
n
ℓ
) 3F2(−ξ,−e, e− n− 1ℓ− n,−ℓ
∣∣∣∣ 1) ∈ R[ξ]. (5.32)
Note that ψℓ,ℓe (ξ) has degree exactly e, from which it follows that
|Sℓ,ℓ(Y )| ⩽ e+ 1. (5.33)
Likewise, we find that {ξ ̸= 0 : 2ξ ∈ Sm,m(Y )} is a set of zeros of the polynomial
ψm,me (ξ) =
e−1∑
r=0
((
n
r
)
−
(
n
r − 1
))
3F2
(
−ξ,−r, r − n− 1
m− n,−m
∣∣∣∣ 1)
+
ωm
(
n−2e
m−e
)((
n
e
)
−
(
n
e−1
))
2n−2eκe
(
n
m
) 3F2(−ξ,−e, e− n− 1m− n,−m
∣∣∣∣ 1) ∈ R[ξ], (5.34)
and hence that
|Sm,m(Y )| ⩽ e+ 1. (5.35)
Finally, by (4.14), (5.12), and (5.27), we have
O =
√
ωℓωm
(
n−2e
ℓ−e
)(
n−2e
m−e
)
2n−2eκe
√
|Yℓ||Ym|
Ĕℓ,me |Yℓ×Ym
=
√
ωℓωm
(
n−m
e
)(
n−2e
ℓ−e
)((
n
e
)
−
(
n
e−1
))
2n−2eκe
√
|Yℓ||Ym|
(
n
ℓ
)(
n−ℓ
e
)
E. Bannai et al.: Tight relative t-designs on two shells in hypercubes, . . . 187
×
min{ℓ,n−m}∑
ξ=0
3F2
(
−ξ,−e, e− n− 1
m− n,−ℓ
∣∣∣∣ 1)A2ξ+m−ℓ|Yℓ×Ym .
Hence it follows that {ξ : 2ξ +m− ℓ ∈ Sℓ,m(Y )} is a set of zeros of the polynomial
ψℓ,me (ξ) = 3F2
(
−ξ,−e, e− n− 1
m− n,−ℓ
∣∣∣∣ 1) ∈ R[ξ], (5.36)
and that
|Sℓ,m(Y )| = |Sm,ℓ(Y )| ⩽ e. (5.37)
By (5.30), (5.33), (5.35), and (5.37), we have
dimH ⩽ 4e+ 2.
Since H ′ is a subspace of H , it follows from (5.28) that H = H ′. In particular, H is a
C-algebra. It is also clear that H is closed under † and contains J|Y |. We now conclude
that H is a coherent algebra. Note also that equality holds in each of (5.33), (5.35), and
(5.37).
To summarize:
Theorem 5.3. Recall Assumption 5.2. With the above notation, the following hold:
(i) The set H from (5.29) is a coherent algebra of type
[
e+1 e
e e+1
]
.
(ii) The sets of zeros of the polynomials ψℓ,ℓe (ξ), ψ
m,m
e (ξ), and ψ
ℓ,m
e (ξ) from (5.32),
(5.34), and (5.36) are given respectively by
{ξ ̸= 0 : 2ξ ∈ Sℓ,ℓ(Y )}, {ξ ̸= 0 : 2ξ ∈ Sm,m(Y )}, and
{ξ : 2ξ +m− ℓ ∈ Sℓ,m(Y )}.
In particular, the zeros of these polynomials are integral.
Concerning the scalars ωℓ and ωm appearing in the polynomials ψℓ,ℓe (ξ) and ψ
m,m
e (ξ),
it follows that
Proposition 5.4. Recall Assumption 5.2. The scalars ωℓ and ωm satisfies
ωm
ωℓ
=
(
n
m
)(
n−2e
ℓ−e
)(
n
ℓ
)(
n−2e
m−e
) · |Ym| − ( ne−1)
|Yℓ| −
(
n
e−1
) .
In particular, the weight function ω is unique up to a scalar multiple.
Proof. By comparing the diagonal entries of both sides in (5.31), we have
1 =
ψℓ,ℓe (0)
|Yℓ|
=
1
|Yℓ|
( n
e− 1
)
+
ωℓ
(
n−2e
ℓ−e
)((
n
e
)
−
(
n
e−1
))
2n−2eκe
(
n
ℓ
)
 .
Likewise,
1 =
ψm,me (0)
|Ym|
=
1
|Ym|
( n
e− 1
)
+
ωm
(
n−2e
m−e
)((
n
e
)
−
(
n
e−1
))
2n−2eκe
(
n
m
)
 .
By eliminating κe, we obtain the formula for ωm(ωℓ)−1. The uniqueness of ω follows from
this and (5.21).
188 Ars Math. Contemp. 22 (2022) #P2.01 / 163–205
Example 5.5. Suppose that e = 1. In this case, Theorem 5.3(i) was previously obtained by
Bannai, Bannai, and Bannai [5, Theorem 2.2 (i)]. Moreover, Theorem 5.3(ii) and Proposi-
tion 5.4 are together equivalent to [5, Proposition 4.3].
Example 5.6. Suppose that e = 2. Then we have
ψℓ,me (ξ) = 1 +
(−ξ)(−2)(1− n)
(m− n)(−ℓ)
+
(−ξ)(1− ξ)(−2)(−1)(1− n)(2− n)
(m− n)(m− n+ 1)(−ℓ)(1− ℓ)2
= 1− 2(n− 1)ξ
(n−m)ℓ
+
(n− 1)(n− 2)ξ(ξ − 1)
(n−m)(n−m− 1)ℓ(ℓ− 1)
.
From Example 3.13 we find two parameter sets satisfying Assumption 5.2:
n ℓ m ξ
22 6 7 3, 5
22 6 15 1, 3
The zeros ξ given in the last column are indeed integers. Note that the other two parameter
sets in Example 3.13 correspond to the complements of these two; cf. Lemma 5.1. On the
other hand, the existence of tight relative 4-designs with the following feasible parameter
sets was left open in [9, Section 6]:
n ℓ m ξ
37 9 16 114 (71±
√
337)
37 9 21 114 (55±
√
337)
41 15 16 126 (237±
√
1569)
41 15 25 126 (153±
√
1569)
Here, we are again taking Lemma 5.1 into account. Observe that the zeros ξ are irrational,
thus proving the non-existence.
We end this section with a comment on the expressions of the polynomials ψℓ,ℓe (ξ)
and ψm,me (ξ). We first invoke the following identity which agrees with the formula of the
backward shift operator on the dual Hahn polynomials (cf. [31, Section 1.6]):
α(N + 1)(α+ β + 2r)Qr(ξ;α− 1, β,N + 1)
= (α+ r)(α+ β + r)(N + 1− r)Qr(ξ − 1;α, β,N)
− r(α+ β +N + 1 + r)(β + r)Qr−1(ξ − 1;α, β,N). (5.38)
This can be routinely verified by writing the LHS as a linear combination of the polynomi-
als (1− ξ)i (0 ⩽ i ⩽ r) using
(−ξ)i = (1− ξ)i − i(1− ξ)i−1,
and then comparing the coefficients of both sides. Setting α = ℓ − n, β = −ℓ − 1, and
N = ℓ − 1 in (5.38), it follows that the first term of the RHS in (5.32) is rewritten as
follows:
e−1∑
r=0
((
n
r
)
−
(
n
r − 1
))
3F2
(
−ξ,−r, r − n− 1
ℓ− n,−ℓ
∣∣∣∣ 1)
E. Bannai et al.: Tight relative t-designs on two shells in hypercubes, . . . 189
=
e−1∑
r=0
n!(n− 2r + 1)
r!(n− r + 1)!
Qr(ξ;α− 1, β,N + 1)
=
n!
ℓ(n− ℓ)
e−1∑
r=0
(
(ℓ− n+ r)(r − n− 1)(ℓ− r)
r!(n− r + 1)!
Qr(ξ − 1;α, β,N)
−r(r + ℓ− n− 1)(r − ℓ− 1)
r!(n− r + 1)!
Qr−1(ξ − 1;α, β,N)
)
=
n!
ℓ(n− ℓ)
· (−1)(ℓ− n+ e− 1)(ℓ− e+ 1)
(e− 1)!(n− e+ 1)!
Qe−1(ξ − 1;α, β,N)
=
(
n
e− 1
)
(n− ℓ− e+ 1)(ℓ− e+ 1)
ℓ(n− ℓ) 3
F2
(
1− ξ, 1− e, e− n− 1
ℓ− n+ 1, 1− ℓ
∣∣∣∣ 1) .
Likewise, the first term of the RHS in (5.34) is given by
e−1∑
r=0
((
n
r
)
−
(
n
r − 1
))
3F2
(
−ξ,−r, r − n− 1
m− n,−m
∣∣∣∣ 1)
=
(
n
e− 1
)
(n−m− e+ 1)(m− e+ 1)
m(n−m) 3
F2
(
1− ξ, 1− e, e− n− 1
m− n+ 1, 1−m
∣∣∣∣ 1) .
6 Zeros of the Hahn and Hermite polynomials
Recall the Hahn polynomials Qr(ξ;α, β,N) from (4.11). Recall also that the zeros of
orthogonal polynomials are always real and simple; see, e.g., [42, Theorem 3.3.1]. It is
well known that we can obtain the Hermite polynomials as limits of the Hahn polynomials;
cf. [30, 31]. In this section, we revisit this limit process and describe the limit behavior of
the zeros of the Qr(ξ;α, β,N), in a special case which is suited to our purpose.
Assumption 6.1. Throughout this section, we assume that α < −N and β < −N , so
that the Qr(ξ;α, β,N) satisfy the orthogonality relation (4.12). We consider the following
limit:
ϵ := − α+ β√
αβN
→ +0.
We write
α =
αϵ
ϵ2
, β =
βϵ
ϵ2
, N =
Nϵ
ϵ2
,
and assume further that
lim
ϵ→+0
Nϵ
αϵ + βϵ
= 0, lim
ϵ→+0
βϵ
αϵ + βϵ
= ρ ∈ [0, 1].
Remark 6.2. We do not require in Assumption 6.1 that αϵ, βϵ, and Nϵ are uniquely deter-
mined by ϵ. In other words, these are multi-valued functions of ϵ in general (for admissible
values of ϵ), but their limit behaviors are uniformly governed by ϵ.
With reference to Assumption 6.1, observe that
lim
ϵ→+0
αϵ = lim
ϵ→+0
αϵ2 = lim
ϵ→+0
αϵ + βϵ
βϵ
· αϵ + βϵ
Nϵ
= −∞.
190 Ars Math. Contemp. 22 (2022) #P2.01 / 163–205
Likewise, we have
lim
ϵ→+0
βϵ = −∞, lim
ϵ→+0
Nϵ =
1
ρ(1− ρ)
∈ [4,∞].
We will work with the normalized (or monic) Hahn polynomials:
qr(ξ) = qr(ξ; ϵ) =
(α+ 1)r(−N)r
(r + α+ β + 1)r
Qr(ξ;α, β,N). (6.1)
Their recurrence relation is given by (cf. [31, Section 1.5])
ξqr(ξ) = qr+1(ξ) + (ar + br)qr(ξ) + ar−1brqr−1(ξ), (6.2)
where q−1(ξ) := 0, and
ar =
(r + α+ β + 1)(r + α+ 1)(N − r)
(2r + α+ β + 1)(2r + α+ β + 2)
,
br =
r(r + α+ β +N + 1)(r + β)
(2r + α+ β)(2r + α+ β + 1)
.
For convenience, let
λϵ =
√
2(αϵ + βϵ +Nϵ)
αϵ + βϵ
.
Note that
lim
ϵ→+0
λϵ =
√
2. (6.3)
Consider the polynomial q̃r(η; ϵ) in the new indeterminate η defined by
q̃r(η) = q̃r(η; ϵ) = qr
(
λϵη
ϵ
+
αϵNϵ
(αϵ + βϵ)ϵ2
)
· ϵ
r
(λϵ)r
∈ R[η].
Note that q̃r(η) is also monic with degree r in η. Then (6.2) becomes
ηq̃r(η) = q̃r+1(η) +
1
λϵ
(
(ar + br)ϵ−
αϵNϵ
(αϵ + βϵ)ϵ
)
q̃r(η) +
ar−1brϵ
2
(λϵ)2
q̃r−1(η). (6.4)
It is a straightforward matter to show that
1
λϵ
(
(ar + br)ϵ−
αϵNϵ
(αϵ + βϵ)ϵ
)
= −(µϵ + rσϵ)ϵ+O(ϵ3), (6.5)
ar−1brϵ
2
(λϵ)2
=
r
2
+O(ϵ2), (6.6)
where
µϵ :=
(αϵ − βϵ)Nϵ
λϵ(αϵ + βϵ)2
, σϵ :=
(αϵ − βϵ)(αϵ + βϵ + 2Nϵ)
λϵ(αϵ + βϵ)2
are convergent:
lim
ϵ→+0
µϵ = 0, lim
ϵ→+0
σϵ =
1− 2ρ√
2
. (6.7)
E. Bannai et al.: Tight relative t-designs on two shells in hypercubes, . . . 191
Recall the Hermite polynomials [31, Section 1.13]
Hr(η) = (2η)
r
2F0
(
−r/2,−(r − 1)/2
−
∣∣∣∣− 1η2
)
∈ R[η] (r = 0, 1, 2, . . .).
Their normalized recurrence relation is given by
ηhr(η) = hr+1(η) +
r
2
hr−1(η), (6.8)
where
hr(η) =
Hr(η)
2r
, (6.9)
and h−1(η) := 0. We also note that
dhr
dη
(η) = rhr−1(η), (6.10)
and that
hr(−η) = (−1)rhr(η). (6.11)
Since q̃0(η) = h0(η) = 1, it follows from (6.4) – (6.8) that
lim
ϵ→+0
q̃r(η; ϵ) = hr(η) (6.12)
in the sense of coefficient-wise convergence.
We now set
q̃r(η; 0) = hr(η),
and discuss partial derivatives of q̃r(η; ϵ) as a bivariate function of η and ϵ. First, it follows
from (6.10) and (6.12) that
lim
ϵ→+0
∂q̃r
∂η
(η; ϵ) =
dhr
dη
(η) = rhr−1(η). (6.13)
Concerning the partial differentiability of q̃r(η; ϵ) with respect to ϵ, it follows that
Lemma 6.3. The function q̃r(η; ϵ) is partially right differentiable with respect to ϵ at (η, 0),
and we have
∂q̃r
∂ϵ
(η; 0) =
r(1− 2ρ)
3
√
2
(
(r − 1 + η2)hr−1(η)− ηhr(η)
)
.
Proof. Throughout the proof, we fix η ∈ R and set
∆r(ϵ) = ∆r(η; ϵ) =
q̃r(η; ϵ)− hr(η)
ϵ
.
It follows from (6.4) – (6.8) and (6.12) that
η∆r(ϵ) = ∆r+1(ϵ)− (µϵ + rσϵ)q̃r(η; ϵ) +
r
2
∆r−1(ϵ) +O(ϵ)
= ∆r+1(ϵ)− rσ0hr(η) +
r
2
∆r−1(ϵ) + o(1), (6.14)
192 Ars Math. Contemp. 22 (2022) #P2.01 / 163–205
where we set
σ0 := lim
ϵ→+0
σϵ =
1− 2ρ√
2
for brevity. Since q̃0(η; ϵ) = 1, we have ∆0(ϵ) = 0. Solving the recurrence (6.14) using
this initial condition and (6.8), we routinely obtain
∆r(ϵ) =
r(r − 1)
2
σ0hr−1(η) +
r(r − 1)(r − 2)
12
σ0hr−3(η) + o(1),
where h−1(η) = h−2(η) = h−3(η) := 0. It follows that q̃r(η; ϵ) is partially right differen-
tiable with respect to ϵ at (η, 0):
∂q̃r
∂ϵ
(η; 0) = lim
ϵ→+0
∆r(ϵ)
=
r(r − 1)
2
σ0hr−1(η) +
r(r − 1)(r − 2)
12
σ0hr−3(η).
Finally, from (6.8) it follows that
∂q̃r
∂ϵ
(η; 0) =
r(r − 1)
2
σ0hr−1(η) +
r(r − 1)
6
σ0
(
ηhr−2(η)− hr−1(η)
)
=
r(r − 1)
3
σ0hr−1(η) +
r
3
σ0η
(
ηhr−1(η)− hr(η)
)
=
rσ0
3
(
(r − 1 + η2)hr−1(η)− ηhr(η)
)
,
as desired.
Proposition 6.4. Recall Assumption 6.1. Fix a positive integer e, and let
ξ−⌊e/2⌋ < · · · < ξ−1 < (ξ0) < ξ1 < · · · < ξ⌊e/2⌋,
η−⌊e/2⌋ < · · · < η−1 < (η0) < η1 < · · · < η⌊e/2⌋
be the zeros of qe(ξ; ϵ) and he(η) from (6.1) and (6.9), respectively, where ξ0 and η0 appear
only when e is odd. Then ξi satisfies
lim
ϵ→+0
(
ξi −
λϵηi
ϵ
− αϵNϵ
(αϵ + βϵ)ϵ2
)
=
2ρ− 1
3
(
e− 1 + (ηi)2
)
as a function of ϵ, for i = −⌊e/2⌋, . . . ,−1, (0), 1, . . . , ⌊e/2⌋.
Proof. Define τi by
ξi =
λϵ(ηi + τi)
ϵ
+
αϵNϵ
(αϵ + βϵ)ϵ2
,
so that ηi + τi is a zero of q̃e(η; ϵ). Then, from (6.12) it follows that
lim
ϵ→+0
τi = 0. (6.15)
For the moment, fix i. Then we have
0 = q̃e(ηi + τi; ϵ) = q̃e(ηi; ϵ) +
∂q̃e
∂η
(ηi + θτi; ϵ)τi
E. Bannai et al.: Tight relative t-designs on two shells in hypercubes, . . . 193
for some θ ∈ (0, 1) depending on ϵ. Hence, from (6.13), (6.15), Lemma 6.3, and since
q̃e(ηi; 0) = he(ηi) = 0,
it follows that
lim
ϵ→+0
τi
ϵ
= − 1
ehe−1(ηi)
lim
ϵ→+0
q̃e(ηi; ϵ)
ϵ
= − 1
ehe−1(ηi)
∂q̃e
∂ϵ
(ηi; 0)
=
2ρ− 1
3
√
2
(
e− 1 + (ηi)2
)
,
where we note that he(η) and he−1(η) have no common zero by the general theory of
orthogonal polynomials; see, e.g., [42, Theorem 3.3.2]. By (6.3), we have
lim
ϵ→+0
(
ξi −
λϵηi
ϵ
− αϵNϵ
(αϵ + βϵ)ϵ2
)
= lim
ϵ→+0
λϵτi
ϵ
=
2ρ− 1
3
(
e− 1 + (ηi)2
)
.
This completes the proof.
The following is part of the estimates on the zeros of he(η) used in [1].3
Proposition 6.5 ([1, Proposition 13]). Fix a positive integer e, and let the ηi be as in
Proposition 6.4. Then η−i = −ηi for all i. Moreover, the following hold:
1. If e is odd and e ⩾ 5, then η0 = 0 and (η1)2 < 3/2.
2. If e is even and e ⩾ 8, then (η2)2 − (η1)2 < 3/2.
Proof. That η−i = −ηi is immediate from (6.11). We now write ηi = ηei to compare these
zeros for different values of e. Then, as an application of Sturm’s method, it follows that
√
2e+ 1 ηei <
√
2e′ + 1 ηe
′
i (i = 1, 2, . . . , ⌊e′/2⌋),
whenever e′ < e and e′ ≡ e (mod 2); see the comments preceding (6.31.19) in [42]. Since
h3(η) = η
3 − 3
2
η, h4(η) = η
4 − 3η2 + 3
4
,
we have
η31 =
√
3
2
, η42 =
√
3 +
√
6
2
.
Hence, for odd e ⩾ 5 we have
(ηe1)
2 <
7
2e+ 1
(η31)
2 =
21
4e+ 2
<
3
2
,
and for even e ⩾ 8 we have
(ηe2)
2 − (ηe1)2 < (ηe2)2 <
9
2e+ 1
(η42)
2 =
27 + 9
√
6
4e+ 2
<
3
2
,
as desired.
3Bannai [1] worked with the polynomial
√
2ehe(η/
√
2). We may remark that the upper bounds
√
3 men-
tioned in Proposition 13 (i) and (ii) in [1] should both be 3. See also [22, Proposition 2.4].
194 Ars Math. Contemp. 22 (2022) #P2.01 / 163–205
7 A finiteness result for tight relative 2e-designs on two shells in Qn
In this section, we prove that
Theorem 7.1. For any δ ∈ (0, 1/2), there exists e0 = e0(δ) > 0 with the property that,
for every given integer e ⩾ e0 and each constant c > 0, there are only finitely many tight
relative 2e-designs (Y, ω) (up to scalar multiples of ω) supported on two shells Xℓ ⊔Xm
in Qn satisfying Assumption 5.2 such that
ℓ < c · nδ. (7.1)
Our proof is an application of Bannai’s method from [1]. We will use the following
result, which is a variation of [40, Satz I]:
Proposition 7.2. For any ϑ > 0 and δ ∈ (0, 1/ϑ), there exists k0 = k0(ϑ, δ) > 0 such
that the following holds for every given integer k ⩾ k0 and each constant c > 0: for all but
finitely many pairs (a, b) of positive integers with
b < c · aδ,
the product of k consecutive odd integers
(2a+ 1)(2a+ 3) · · · (2a+ 2k− 1)
has a prime factor which is greater than 2k + 1 and whose exponent in this product is
greater than that in
(b+ 1)(b+ 2) · · · (b+ ⌊ϑk⌋).
The proof of Proposition 7.2 will be deferred to the appendix.
We will establish Theorem 7.1 by contradiction:
Assumption 7.3. We fix δ ∈ (0, 1/2). Let k0 = k0(2, δ) > 0 be as in Proposition 7.2
(applied to ϑ = 2), and set
e0 = e0(δ) = max{2k0, 8}.
We also fix a positive integer e ⩾ e0 and a constant c > 0. Throughout the proof, we
assume that there exist infinitely many tight relative 2e-designs (Y, ω) in question.
Let Θ denote the set of triples (ℓ,m, n) ∈ N3 taken by those (Y, ω) in Assumption 7.3.
Recall from Proposition 5.4 that ω is uniquely determined by Y up to a scalar multiple.
Moreover, for each (ℓ,m, n) ∈ Θ there are only finitely many choices for Y . Hence we
have
|Θ| =∞. (7.2)
For the moment, we fix (ℓ,m, n) ∈ Θ and consider the polynomial ψℓ,me (ξ) (which
also depends on n) from (5.36). We recall that
ψℓ,me (ξ) = Qe(ξ;α, β,N),
where
α = m− n− 1, β = −m− 1, N = ℓ. (7.3)
E. Bannai et al.: Tight relative t-designs on two shells in hypercubes, . . . 195
We note that α, β < −N in view of Assumption 5.2. By Theorem 5.3(ii), if we let
ξ−⌊e/2⌋ < · · · < ξ−1 < (ξ0) < ξ1 < · · · < ξ⌊e/2⌋ (7.4)
denote the zeros of ψℓ,me (ξ) (cf. Proposition 6.4), then we have
ξi ∈ {0, 1, . . . , ℓ} for all i. (7.5)
We also rewrite ψℓ,me (ξ) as follows:
ψℓ,me (ξ) =
e∑
i=0
se−i(−1)i(−ξ)i,
where
se−i =
(
e
i
)
(e− n− 1)i
(m− n)i(−ℓ)i
(0 ⩽ i ⩽ e).
From (7.5) it follows that the polynomial ψℓ,me (ξ)/s0 is monic and integral:
ψℓ,me (ξ)
s0
=
e∑
i=0
se−i
s0
(−1)i(−ξ)i = (ξ − ξ−⌊e/2⌋) · · · (ξ − ξ⌊e/2⌋) ∈ Z[ξ], (7.6)
where the factor (ξ − ξ0) appears only when e is odd. Since (−1)i(−ξ)i is also monic and
integral, and has degree i for 0 ⩽ i ⩽ e, it follows that
si
s0
= (−1)i
(
e
i
)
(n−m− e+ 1)i(ℓ− e+ 1)i
(n− 2e+ 2)i
∈ Z\{0} (0 ⩽ i ⩽ e), (7.7)
where that these coefficients are non-zero follows from Assumption 5.2.
We now consider the map f : Θ→ [0, 1]2 defined by
f(ℓ,m, n) =
(
ℓ
n
,
m
n
)
∈ [0, 1]2 ((ℓ,m, n) ∈ Θ).
Recall (7.2). Moreover, from (7.1) it follows that
|f−1(a, b)| <∞ ((a, b) ∈ [0, 1]2). (7.8)
Hence it follows that
|f(Θ)| =∞,
so that f(Θ) has at least one accumulation point in [0, 1]2. Again by (7.1), such an accu-
mulation point must be of the form
(0, ρ) ∈ [0, 1]2.
We next show that the parameters α, β, and N from (7.3) satisfy Assumption 6.1 when
f(ℓ,m, n)→ (0, ρ).
Claim 7.4. ℓ,m, n−m→∞ as f(ℓ,m, n)→ (0, ρ).
196 Ars Math. Contemp. 22 (2022) #P2.01 / 163–205
Proof. Since m,n −m ⩾ ℓ by Assumption 5.2, it suffices to show that ℓ → ∞. Suppose
the contrary, i.e., that there is a sequence (ℓk,mk, nk) (k ∈ N) of distinct elements of Θ
such that
lim
k→∞
f(ℓk,mk, nk) = (0, ρ), sup
k
ℓk <∞.
Since the ℓk are bounded, it follows from (7.5) and (7.6) that there are only finitely many
choices for ψℓ,me (ξ)/s0 when (ℓ,m, n) ranges over this sequence. In particular, there are
only finitely many choices for each of the coefficients s1/s0 and s2/s0, and hence the same
is true (cf. (7.7)) for each of
n−m− e+ 1
n− 2e+ 2
,
n−m− e+ 2
n− 2e+ 3
.
However, it is immediate to see that these distinct scalars in turn determine n and m
uniquely, from which it follows that the nk are bounded, a contradiction.
Claim 7.5. ℓm(n−m)/n2 →∞ as f(ℓ,m, n)→ (0, ρ).
Proof. If 0 < ρ < 1 then the result follows from Claim 7.4 and since
m(n−m)
n2
→ ρ(1− ρ) > 0.
Suppose next that ρ = 1. Suppose moreover that there is a sequence (ℓk,mk, nk)
(k ∈ N) of distinct elements of Θ such that
lim
k→∞
f(ℓk,mk, nk) = (0, 1), sup
k
ℓkmk(nk −mk)
(nk)2
<∞.
Since mk/nk → 1, we then have
sup
k
ℓk(nk −mk)
nk
<∞.
Let
rk =
(nk −mk − e+ 1)(ℓk − e+ 1)
nk − 2e+ 2
, tk =
(nk −mk − e+ 2)(ℓk − e+ 2)
nk − 2e+ 3
.
Then the rk and the tk are bounded since
rk ≈ tk ≈
ℓk(nk −mk)
nk
by Claim 7.4. From (7.7) it follows that s1/s0 and s2/s0 are bounded as well, and hence
take only finitely many non-zero integral values when (ℓ,m, n) ranges over this sequence.
It follows that the rk and the tk can assume only finitely many values, and then since
rk ≈ tk we must have rk = tk for sufficiently large k. However, it is again immediate to
see that rk ̸= tk for every k ∈ N, and hence this is absurd. It follows that the result holds
when ρ = 1.
Finally, suppose that ρ = 0. For every (ℓ,m, n) ∈ Θ we have
e
(m− e+ 1)(ℓ− e+ 1)
n− 2e+ 2
=
s1
s0
+ e(ℓ− e+ 1),
E. Bannai et al.: Tight relative t-designs on two shells in hypercubes, . . . 197
(
e
2
)
(m− e+ 1)2(ℓ− e+ 1)2
(n− 2e+ 2)2
=
s2
s0
+ (e− 1)(ℓ− e+ 2)s1
s0
+
(
e
2
)
(ℓ− e+ 1)2.
From (7.7) and Assumption 5.2 it follows that these scalars are non-zero integers. By the
same argument as above, but working with these two scalars instead of s1/s0 and s2/s0,
we conclude that the result holds in this case as well.
By Claims 7.4 and 7.5, it follows that the parameters α, β, and N from (7.3) satisfy
Assumption 6.1 when f(ℓ,m, n)→ (0, ρ), since
− α+ β√
αβN
≈ n√
ℓm(n−m)
,
N
α+ β
≈ − ℓ
n
,
β
α+ β
≈ m
n
.
Note that the scalar ρ in Assumption 6.1 agrees with the one used here in this case. Hence
we are now in the position to apply the results of the previous section to ψℓ,me (ξ), which is
the Hahn polynomial having these parameters.
Claim 7.6. We have ρ = 1/2. In particular, (0, 1/2) is a unique accumulation point of
f(Θ). Moreover, we have n = 2m for all but finitely many (ℓ,m, n) ∈ Θ.
Proof. Let the ξi be as in (7.4). Then from Propositions 6.4 and 6.5 it follows that
ξi + ξ−i − ξj − ξ−j →
4ρ− 2
3
(
(ηi)
2 − (ηj)2
)
for all i, j, (7.9)
as f(ℓ,m, n)→ (0, ρ), where the ηi are the zeros of the monic Hermite polynomial he(η)
from (6.9) as in Proposition 6.4. Recall that e ⩾ 8 by Assumption 7.3. Set (i, j) = (1, 0)
in (7.9) if e is odd, and (i, j) = (2, 1) if e is even. Then, since∣∣∣∣4ρ− 23
∣∣∣∣ ⩽ 23 ,
it follows from Proposition 6.5 that the RHS in (7.9) lies in the open interval (−1, 1).
However, the LHS in (7.9) is always an integer by (7.5), so that this is possible only when
the RHS equals zero, i.e., ρ = 1/2. In particular, we have shown that (0, 1/2) is a unique
accumulation point of f(Θ).
Again by (7.5) and (7.9), we then have
ξi + ξ−i = ξj + ξ−j for all i, j,
provided that f(ℓ,m, n) is sufficiently close to (0, 1/2). By the uniqueness of the accu-
mulation point and (7.8), this last condition on f(ℓ,m, n) can be rephrased as “for all but
finitely many (ℓ,m, n) ∈ Θ.” Now, let ξ̃ be the average of the zeros ξi of ψℓ,me (ξ). Then
the above identity means that the ξi are symmetric with respect to ξ̃. Hence, if we write
ψℓ,me (ξ)
s0
=
e∑
i=0
we−i(ξ − ξ̃)i,
then we have
w2i−1 = 0 (1 ⩽ i ⩽ ⌈e/2⌉)
198 Ars Math. Contemp. 22 (2022) #P2.01 / 163–205
for all but finitely many (ℓ,m, n) ∈ Θ. On the other hand, using (7.6) and (7.7), we
routinely obtain
w3 =
(
e
3
)
(n− 2ℓ)(n− 2m)
× (ℓ− e+ 1)(m− e+ 1)(n− ℓ− e+ 1)(n−m− e+ 1)
(n− 2e+ 2)3(n− 2e+ 3)(n− 2e+ 4)
.
Hence, by Assumption 5.2, that w3 = 0 forces n = 2m. The claim is proved.
By virtue of Claim 7.6, we may now assume without loss of generality that
n = 2m ((ℓ,m, n) ∈ Θ),
by discarding a finite number of exceptions. Set
k =
⌊e
2
⌋
,
and let c′ be a constant such that c′ > 2δc. Note that
k ⩾ k0 = k0(2, δ)
by Assumption 7.3. Let (ℓ,m, 2m) ∈ Θ. We have
c · (2m)δ < c′ · (m− e+ 1)δ
provided that m is large. Hence it follows from Proposition 7.2 (applied to ϑ = 2) and
(7.1) that if m is sufficiently large then there is a prime p > 2k+ 1 such that
νp((2m− 2e+ 3)(2m− 2e+ 5) · · · (2m− 2e+ 2k+ 1)) > νp((ℓ− e+ 1)2k),
where νp(n) denotes the exponent of p in n. Assuming that this is the case, let i (1 ⩽ i ⩽ k)
be such that
νp(2m− 2e+ 2i+ 1) > 0.
Observe that i is unique since p > 2k+ 1, so that we have
νp(2m− 2e+ 2i+ 1) > νp((ℓ− e+ 1)2k).
Moreover, we have
gcd(2m− 2e+ 2i+ 1,m− e+ i+ j) = gcd(2j − 1,m− e+ i+ j) < p
for 1 ⩽ j ⩽ i, from which it follows that
νp((m− e+ i+ 1)i) = 0.
By these comments and since
2i ⩽ e < p,
it follows from (7.7) (with n = 2m) that
νp
(
s2i
s0
)
= νp
(
(m− e+ 1)2i(ℓ− e+ 1)2i
(2m− 2e+ 2)2i
)
E. Bannai et al.: Tight relative t-designs on two shells in hypercubes, . . . 199
= νp
(
(m− e+ i+ 1)i(ℓ− e+ 1)2i
2i(2m− 2e+ 3)(2m− 2e+ 5) · · · (2m− 2e+ 2i+ 1)
)
< 0.
However, this contradicts the fact that s2i/s0 is a non-zero integer. Hence we now conclude
that Θ must be finite.
The proof of Theorem 7.1 is complete.
ORCID iDs
Hajime Tanaka https://orcid.org/0000-0002-5958-0375
Yan Zhu https://orcid.org/0000-0001-7164-9198
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Appendix A Proof of Proposition 7.2
Our proof of Proposition 7.2 is a slight modification of (the first part of) that of [40, Satz I].
For a positive integer n, let
χn = 1 · 3 · 5 · · · (2n− 1) =
(2n)!
2nn!
.
Observe that the exponent νp(χn) of an odd prime p in χn is given by
νp(χn) =
⌊logp(2n)⌋∑
i=1
(⌊
2n
pi
⌋
−
⌊
n
pi
⌋)
=
⌊logp(2n)⌋∑
i=1
⌊
n
pi
+
1
2
⌋
, (A.1)
where we have used
⌊ξ⌋+
⌊
ξ +
1
2
⌋
= ⌊2ξ⌋ (ξ ∈ R).
Now, let (a, b) be a pair of positive integers with
b < c · aδ, (A.2)
which does not satisfy the desired property about a prime factor; in other words,
νp(χa+k)− νp(χa) ⩽ νp((b+ ⌊ϑk⌋)!)− νp(b!) if p > 2k+ 1. (A.3)
Our aim is to show that a is bounded in terms of ϑ, δ, c, and k, and hence so is b by (A.2),
from which it follows that there are only finitely many such pairs. (We will specify k0 =
k0(ϑ, δ) at the end of the proof.) To this end, we may assume for example that
a > k, c · aδ > k+ 1. (A.4)
Without loss of generality, we may also assume that
b > k, (A.5)
for otherwise the pair (a, k+ 1) would also satisfy (A.2) and (A.3).
Let
s =
χa+k
χkχa
· b!
(b+ ⌊ϑk⌋)!
.
Then from (A.3) it follows that
s ⩽
∏
3⩽p⩽2k+1
pνp(s), (A.6)
where the product in the RHS is over the odd primes p ⩽ 2k+ 1, and where
νp(s) = νp(χa+k)− νp(χk)− νp(χa) + νp(b!)− νp((b+ ⌊ϑk⌋)!).
By (A.1), for every odd prime p we have
νp(s) ⩽ νp(χa+k)− νp(χk)− νp(χa)
204 Ars Math. Contemp. 22 (2022) #P2.01 / 163–205
=
⌊logp(2a+2k)⌋∑
i=1
(⌊
a+ k
pi
+
1
2
⌋
−
⌊
k
pi
+
1
2
⌋
−
⌊
a
pi
+
1
2
⌋)
.
Note that ⌊
ξ + η +
1
2
⌋
−
⌊
ξ +
1
2
⌋
−
⌊
η +
1
2
⌋
∈ {−1, 0, 1} (ξ, η ∈ R).
Hence it follows that
νp(s) ⩽ logp(2a+ 2k) =
ln(2a+ 2k)
ln p
(A.7)
for every odd prime p. From (A.6) and (A.7) it follows that
ln s ⩽ (π(2k+ 1)− 1) ln(2a+ 2k), (A.8)
where π(n) denotes the number of primes at most n.
On the other hand, we have
s =
(2a+ 2k)!k!a!
(a+ k)!(2k)!(2a)!
· b!
(b+ ⌊ϑk⌋)!
.
Using Stirling’s formula
ln(n!) =
(
n+
1
2
)
ln n− n+ ln 2π
2
+ rn,
where
0 < rn <
1
12n
,
we obtain
ln s > (a+ k) ln(a+ k)− k ln k− a ln a+ ϑk− 2
+
(
b+
1
2
)
ln b−
(
b+ ϑk+
1
2
)
ln(b+ ϑk). (A.9)
Let
ã =
a
k
, b̃ =
b
k
.
Note that
ã, b̃ > 1, (A.10)
in view of (A.4) and (A.5). With this notation, we have
ln s > k ln ã+ (ã+ 1)k ln
(
1 +
1
ã
)
− ϑk ln k+ ϑk− 2
− ϑk ln b̃−
(
(b̃+ ϑ)k+
1
2
)
ln
(
1 +
ϑ
b̃
)
> k ln ã− ϑk ln k− 2− ϑk ln b̃−
(
ϑk+
1
2
)
ln
(
1 +
ϑ
b̃
)
E. Bannai et al.: Tight relative t-designs on two shells in hypercubes, . . . 205
> (1− ϑδ)k ln ã− ϑk ln c− ϑδk ln k− 2−
(
ϑk+
1
2
)
ln(1 + ϑ), (A.11)
where the first inequality is a restatement of (A.9), the second follows from
0 < ln(1 + ξ) < ξ (ξ > 0),
and the last one follows from (A.2) and (A.10).
Concerning the prime-counting function π(n), it is known that [37, (3.6)]
π(n) < 1.25506
n
ln n
(n > 1).
By this, (A.8), and (A.10), we have
ln s ⩽ (π(2k+ 1)− 1)
(
ln ã+ ln 2k
(
1 +
1
ã
))
<
(
1.25506
2k+ 1
ln(2k+ 1)
− 1
)
(ln ã+ ln 4k). (A.12)
Combining (A.11) and (A.12), it follows that(
(1− ϑδ)k− 1.25506 2k+ 1
ln(2k+ 1)
+ 1
)
ln ã
< ϑk ln c+ ϑδk ln k+ 2 +
(
ϑk+
1
2
)
ln(1 + ϑ)
+
(
1.25506
2k+ 1
ln(2k+ 1)
− 1
)
ln 4k. (A.13)
If we set
k0 = k0(ϑ, δ) =
1
2
(
exp
(
2.51012
1− ϑδ
)
− 1
)
> 0
for example, then we have
(1− ϑδ)k− 1.25506 2k+ 1
ln(2k+ 1)
+ 1 ⩾
1 + ϑδ
2
> 0 (k ⩾ k0).
Hence, whenever k ⩾ k0, it follows from (A.13) that ln a = ln ã+ ln k is bounded in terms
of ϑ, δ, c, and k, from which and (A.2) it follows that there are only finitely many choices
for the pairs (a, b).
This completes the proof of Proposition 7.2.