¿^creative ^commor
ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 18 (2020) 163-177
https://doi.org/10.26493/1855-3974.1907.3c2
(Also available at http://amc-journal.eu)
Hypergeometric degenerate Bernoulli polynomials and numbers
Takao Komatsu * ©
Department of Mathematical Sciences, School of Science, Zhejiang Sci-Tech University, Hangzhou 310018, China
Received 11 January 2019, accepted 18 February 2020, published online 15 October 2020 Abstract
Carlitz defined the degenerate Bernoulli polynomials ,0„(A, x) by means of the generat-ingfunction t((1 + At)1/A -1) X(1 + At)x/A. In 1875, Glaisher gave several interesting determinant expressions of numbers, including Bernoulli, Cauchy and Euler numbers. In this paper, we show some expressions and properties of hypergeometric degenerate Bernoulli polynomials ^N,„(A, x) and numbers, in particular, in terms of determinants.
The coefficients of the polynomial ^„(A, 0) were completely determined by Howard in 1996. We determine the coefficients of the polynomial Pn,„(A, 0). Hypergeometric Bernoulli numbers and hypergeometric Cauchy numbers appear in the coefficients.
Keywords: Bernoulli numbers, hypergeometric Bernoulli numbers, hypergeometric Cauchy numbers, hypergeometric functions, degenerate Bernoulli numbers, determinants, recurrence relations. Math. Subj. Class. (2020): 11B68, 11B37, 11C20, 15A15, 33C15
1 Introduction
Carlitz [7, 8] defined the degenerate Bernoulli polynomials ,0„(A, x) by means of the generating function
U + At)!A - 0(1 + At)x/A = E ^x) n.
When A ^ 0 in (1.1), Bn(x) = ^„(0, x) are the ordinary Bernoulli polynomials because
/ t \ tfxt t„
]im -*-) (1 + At)x/A =	= V B„(x)- .
A-^ol^ (1 + At)1/A - 1J( + ) et - 1 ^ n( ) n!
* The author thanks the anonymous referee for careful reading of the manuscript and helpful comments and suggestions.
E-mail address: komatsu@zstu.edu.cn (Takao Komatsu) ©® This work is licensed under https://creativecommons.org/licenses/by/4.0/
ARS MATHEMATICA CONTEMPORANEA
106
Ars Math. Contemp. 18 (2020) 105-115
When A ^ 0 and x = 0 in (1.1), Bn defined by
t
P„(0,0) are the classical Bernoulli numbers
é - 1
__+ n
E Bn — .
Z—t — !
(1.2)
The degenerate Bernoulli polynomials in A and x have rational coefficients. When x = 0, Pn(A) = Pn(A, 0) are called degenerate Bernoulli numbers. In [16], explicit formulas for the coefficients of the polynomial P„(A) are found. In [26], a general symmetric identity involving the degenerate Bernoulli polynomials and the sums of generalized falling factorials are proved.
In another direction, hypergeometric Bernoulli polynomials BN,n(x) (see, e.g., [ ]) are defined by the generating function
iFi(1; N +1; t)
t
EBw,n(x) , -!
(1.3)
where 1F1 (a; b; z) is the confluent hypergeometric function defined by
(a)(n) zn
iFi(a;b;z) = E
n=0
(6)(n)
with the rising factorial (x)(n) = x(x +1) • • • (x + n — 1) (n > 1) and (x)(0) = 1. When x = 0 in (1.3), BN,n = BN,n(0) are the hypergeometric Bernoulli numbers ([ , 1 , 14, 15, 19 ]). When N = 1 in (1.3), Bn(x) = B1jn(x) are the ordinary Bernoulli polynomials. When x = 0 and N =1 in (1.3), Bn = B1n(0) are the classical Bernoulli numbers.
Many kinds of generalizations of the Bernoulli numbers have been considered by many authors. For example, such generalizations include poly-Bernoulli numbers, Apostol Bernoulli numbers, various types of q-Bernoulli numbers, Bernoulli Carlitz numbers. One of the advantages of hypergeometric numbers is the natural extension of determinant expressions of the numbers.
A determinant expression of hypergeometric Bernoulli numbers ([2, 18]) is given by
Bn, n = (-1)n—!
N !
(N+1)!	1
N!	N!
0
(N+2)!
N!
(N +1)!
N!
(N+n-1)! (N+n-2)! N!	N!
0
(N +1)!	1
N!	N!
1
N!
(N+n)!
(N+n-1)!
(N+2)! (N+1)!
(1.4)
The determinant expression for the classical Bernoulli numbers Bn = B1n was discovered by Glaisher ([11, p. 53]).
In this paper, we introduce and study the hypergeometric degenerate Bernoulli numbers as total generalizations of degenerate Bernoulli numbers and hypergeometric Bernoulli numbers in the aspects of determinants. By applying Trudi's formula and the inversion formula, we show several arithmetical and combinatorial identities. The coefficients of the polynomial pn(A) were completely determined by Howard in 1996. We determine the coefficients of the polynomial pN,n(A). The constant term and the leading coefficient are exactly equal to Hypergeometric Bernoulli numbers and hypergeometric Cauchy numbers, respectively.
tx
e
-!
T. Komatsu: Hypergeometric degenerate Bernoulli polynomials and numbers	165
2 Definition and preliminary results
Denote the generalized falling factorial by for n > 1
(x|a)n = x(x — a)(x — 2a) • • • (x — (n — 1)a)
with (x|a)0 = 1. When a = 1, (x)n = (x|1)n is the ordinary falling factorial. Define
hypergeometric degenerate Bernoulli polynomials ,0N,n(A, x) by
1	W	tn
1,N — a; N +1; —At J J (1 + At)x/A = ^N,n(A,x)- , (2.1)
' '	n=0	n'
where 2F1 (a, b; c; z) is the Gauss hypergeometric function defined by
(a)(n)(b)(n) zn
2Fi(a;b;z) = E
n=0
(c)(n)
'
When x = 0 in (2.1), ^N,n(A) = ^N,n(A, 0) are the hypergeometric degenerate Bernoulli numbers. When A ^ 0, BN,n (x) = lim^0 ^N,n(A, x) are the hypergeometric Bernoulli polynomials in (1.3). Since
(1 + At)1/A — 1
(1 — AJA)n-1 tn'
in (1.1), we can write
1
2F1 ( 1,N — A; N +1; —At
(1 — A|A)n-1 tN N' t
E (1 — A|A)n-1 tn
\n=N
1 + ^ (1 — A|A)n+n-lN' tn 1 + (1 _ — - - ^
1 + E
(1 — A|A)n-i(N + n)!
(1 — NA|A)n
(N + n)n '
(2.2)
When N =1, ^n(A, x) = (A, x) are degenerate Bernoulli polynomials, defined by
-i
1 + E ^AATt^ (1+ At)x/A ^/n(A,x)nj.
n=0
When N =1 and A ^ 0, Bn(x) = lim^^0 ^N,1(A, x) are the classical Bernoulli polynomials, defined by
-1
1 + E
tn
(n +1)!
ext =
tn
EBn(x) .
n=1	n=0
The definition (2.1) may be obvious or artificial for the readers with different backgrounds. One of our motivations is mentioned in Section 5.
1
t
1
166
ArsMath. Contemp. 18(2020) 163-177
We have the following recurrence relation of hypergeometric degenerate Bernoulli numbers pN,n(A).
Proposition 2.1. For N, n > 1, we have
^ n!(1 - NA|A)w-fcN! _ (A) ,n(A) = - ^ (N + n - k)!k!	(A)
k=0
with PN,0(A) = 1.
Proof: By (2.1) with (2.2), we get
/ ^ (1 - NA|A)iN!^ p (A)tn
1=l1 + £ (" + <)! tJLEo "N,n(A) n

^ « (A)tn + ^ n- (1 - NA|A)n-kN! $N,k(A).
= 2. «N,n (A) ni^^Z. (n + n - k)! kT~ '
n=0	n=1 k=0
Comparing the coefficients on both sides, we obtain for n > 1
«N,n(A) , ^^ (1 - NA|A)n-kN! «N,k(A)
n!
__Y^ (1 - NA| A)n-k N ! p N,k (A) = 0
! ^ (N + n - k)! k! = '	D
k=0
We can use Proposition 2.1 to give an explicit expression for flN,n(A). Theorem 2.2. For N, n > 1,
(A)= n!V( N-)k V (1 - NAlA)ii (1 - NAIA)» ^(A) = n!k=(!)	(NW
(2.3)
n
>i

Remark 2.3. When A ^ 0, Theorem 2.2 is reduced to
n	( N!) k
i1,...,ik>l
as seen in [ , 18]. When A ^ 0 and N =1, there is a combinatorial interpretation of Bernoulli numbers in terms of the cardinality of Z2-graded groupoids [4, Corollary 45].
Proof of Theorem 2.2. The proof is by induction on n. From Proposition 2.1 with n = 1,
(1 - NA)N! p (A)= N!(1 - NA)
«A) = - (n +1)! pn,°(a) = - (N +1)! '
This matches the expression (2.1) when n =1. Assume that the result is valid up to n - 1. For simplicity, put
S	^ (1 - NA|A)ii (1 - NA|A)jfc
Sk (n)= ^-----(N^T '	(2^
1
T. Komatsu: Hypergeometric degenerate Bernoulli polynomials and numbers	167
Then by Proposition 2.1
*-1
ftw,*(A)
r!
-E
(1 - NA|A)*_,N! ftw,i(A) (N + n - 1)! 1!


,=0
(1 - WA|A,"N!- g(1 -N/?!A)n_),N! É (—n !)k s.«)
- £(-N!)k£ " -N,A'A>"_f 'S.(1)


	(N + n)!
(1	- NA|A)*N!
	(N + n)!
(1	- NA|A)*N!
	(N + n)!
(1	- NA|A)*N!
k=1
i = fc
(N + n - 1)!
- Z(-N!)'_' Z (1 - NA|A)*_,IN!SWO
k=2
, = fc- 1
(N + n - 1)!
(N + n)! *
= Z(-N!)kS.(n).
+ Z(-N !)k S.(n)
.=2
k=1
Here, we put n - 1 = ik in the second last equation.
□
There is an alternative form of ftN,*(A) using binomial coefficients. The proof may be similar to that of Theorem 2.2, but a different proof is given.
Theorem 2.4. For N, n > 1,
£n,*(A) = n! Z(-N!)k
.=1
(1 - NA|A)i! (1 - NA|A)i
n V -
k -1+f+Îfc=„ (N + i1)! (N + i.)! •
Proof. Put
1 + w = 2Fi ( 1, N - 1; N +1; -At
By the definition (2.1) with x = 0, we have
£n,*(a) = dt* (1 + w)_
*l
l=0 .=0
	d*
t=0	= dt*
	d* /
k	dt* V2
l=0
1
A;
= > — (-w) ^ dt* V 7 t=0 '=0
.
t=0
t=0
By (2.2), we get
d* 1 — 2F\ 1,N - -; N +1; -At
dt* V2 H A '
= d* (1 - NA|A),,
t=0 dt* VÉ0 (N +0« ,
t=0
= n!Rfc (n).
>0

*
1
w
.
.
.
106
Ars Math. Contemp. 18 (2020) 105-115
where
R () v- (1 - NA|A)i! (1 - NA|A)i Rk (n) =
i1 + ... + ifc (N + ii)i! (N + ik )i ¿, ,...,ifc>0
Thus, we have
i
Pn,U (A)= ]T ]T (-1)k( k)n!Rk (n)
l=0 k=0 ^ '
n	n / j
= n! j(-1)kRk(n) £ U
k=0 l=k ^
= », g(-1)kRk(.)(n +;)
= n' Ê<-1)k (I + >M
n, £(-N'j^n	^ (1 - NA|A),i (1 - NA|A)j
k=1 v 7 ii + ---+ifc
I +V „(N + i 1 )' (N + ik)' • □
3 Hypergeometric degenerate Bernoulli polynomials
In this section, a relation between hypergeometric degenerate Bernoulli polynomials and numbers and some more related properties are shown.
Theorem 3.1. For N > 1 and n > 0,
^w,n(A, x + y) = ["kj (y|A)n-fc @N,k (A, x). k=0 ^ '
Proof. By the definition in (2.1),
V^N,n(A,X + y) — n!
n=0
= (Vi ( 1,N - A; N + 1;-At)) (1 + At)(x+y)/A = (e ^N,n(A,x+y) n.) (y/A)(At)^
( œ	tn\ / œ	tl\
= j 0n, n(A,x + y) - i(y|A)i -\n=0	V \l=0	/
œ / n / )	\ n
E E J (y|A)n-k^N,k (A,x) - .
n=0 k=0
Comparing the coefficients on both sides, we get the desired result.	□
0

T. Komatsu: Hypergeometric degenerate Bernoulli polynomials and numbers	169
By specializing y = 0 in Theorem 3.1, we have a relation between the hypergeometric degenerate Bernoulli polynomials and numbers.
Corollary 3.2. For N > 1 and n > 0,
= Kl (x|A)„-fc^N,fc (A) . k=0 ^ '
Theorem 3.3. For N > 1 and n > 0,
d	n 1(—A)"-fc-1n!
-^,n(A,x) = £ (n- '
k=0 v '
^N,k (A, x)
Proof. By the definition in (2.1),
d œ tn dx	n!
n=0
(2fI( 1,n - A;N +1;—At))-1 dX (1 + At)(x)/A
tn
= log(1 + At)1/A V ^N,n(A,x) -
n!
n=0
A £ (-1)1-1 (At)1
A
\	i=1
œ	/ n— 1
: :
n=1	\k=0
l
tn
X^N,n(A x) -f
n=0
(-A)
n — k — 1
(n — k)k!
(A,x) tn .
Comparing the coefficients on both sides, we get the desired result.
□
4 A determinant expression of hypergeometric degenerate Bernoulli numbers
Theorem 4.1. For N, n > 1, we have
^W,n(A)
( —1)n n!
(1 — WA)W !
(N +1)! (1 —NA|A)2N ! (N+2)!
(1 —NA)N ! (N +1)!
(1 —NA|A)n-iN !	(1 —NA|A)n-2N !
(N+n —1)!	(N+n—2)!
(1 —NA|A)nN !	(1 — NA|A)n-iN !
(N+n)!	(N+n—1)!
1
(1 —NA)N !	,
(N +1)!	1
(1 —NA|A)2N ! (1 —NA)N !
(N+2)!
(N+1)!
1
0
106
Ars Math. Contemp. 18 (2020) 105-115
Remark 4.2. When A ^ 0 in Theorem 4.1, we get a determinant expression of hypergeo-metric Bernoulli numbers B«,n in (1.4). If A ^ 0 and N = 1 in Theorem 4.1, we recover the classical determinant expression of the Bernoulli numbers Bn ([11, p. 53]).
Proof of Theorem 4.1. For simplicity, we put /«,„ = (-1)n/N,n(A)/n! and
(1 - NAlA)i-j+iN! if f(. , = . (N + i - j + 1)! if * " j; f (i ,j) = ) 1 if i = j - 1;
otherwise
and shall prove that
/«,n = |f (* , j)|l<i,j<n .	(4.1)
From Proposition 2.1, we have
n-1 (-1)n-m-1(1 - NA|A)„-mN!
ô (-1) (1 - NA|A)n-mN ! ô PN,n = > -T^r--rj-f1«,.
( N + n — m )!
m=0	v	'
n-1
= E (-1)n-m-1f (n - m , 1)/?«,m .
(4.2)
. =0
When n = 1, it is trivial because by Theorem 2.2
(1 - NA)N!


(N +1)! •
Assume that (4.1) is valid up to n - 1. By expanding along the first row, the right-hand side of (4.1) is equal to
f (1, 1)/«,n-1 -
f (2,1) 1 0 f (3,1)	1
: .... 1 0
f (n - 1, 1) f (n - 1, 3) ••• f (n - 1,n - 1) 1 f (n 1) f (n, 3) ••• f (n,n - 1) f (n,n)
f(n - 1, 1) 1
f(n, 1) f(n, n)
= f (1, 1)/v,n-1 - f (2, 1)/?N,n-2 + ■ ■ ■ + (-1)"-n-1
= £ (-1)n-m-1f (n - m, 1)/SN,m = /?N,n .
m=0
Here, we used the relation (4.2) with /3N 0 = 1.	□
5 Applications of Trudi's formula and inversion relations
One motivation of this paper comes from a 1989 paper of Cameron [6], in which he considered the operator A defined on the set of sequences of non-negative integers as follows: for x = |i„}„>i and z = {z„}„>1, set Ax = z, where
w	/	w	\ -1
1 + £ Z„t" = 1 - £ xntn .	(5.1)
n=1	\ n=1	J
T. Komatsu: Hypergeometric degenerate Bernoulli polynomials and numbers	171
Suppose that x enumerates a class C. Then Ax enumerates the class of disjoint unions of members of C, where the order of the "component" members of C is significant. The operator A also plays an important role for free associative (non-commutative) algebras. More motivations and background together with many concrete examples (in particular, in the aspects of graph theory) by this operator can be seen in [6].
Though only nonnegative numbers in the sequence are treated with combinatorial interpretations in [6], the transformation in (5.1) can be extended to negative or rational numbers too. Some combinatorial interpretations for rational numbers can be found in [3,4], where a categorical setting is proposed. In the sense of Cameron's operator A, we have the following relations.
A -
1
A -
A -
(n +1)! 1
(N + n)n (1 - A|A)n
A
(n +1)! (1 - NA|A)n (N + n)n
Bn
n!
Bn,U n!
fin (A)
n!
fiw,n(A) n!
These relations are interchangeable in the sense of determinants too.
We shall use Trudi's formula to obtain different explicit expressions and inversion relations for the numbers fiN,n(j).
Lemma 5.1. For n > 1, we have
ai a2
an-i
ao ai
an-i
'.. 0
ai ao a2 ai
E
Î1+2Î2+-----+nt„=n
ti + ••• + tn tl,...,tn
(-ao)
n —ti -
aii a22 • • • an
where +in) = ^l+u.+V^ are the multinomial coefficients.
This relation is known as Trudi's formula [ , Vol. 3, p. 214], [ 25] and the case a0 = 1 of this formula is known as Brioschi's formula [ ], [ , Vol. 3, pp. 208-209].
In addition, there exists an inversion formula (see, e.g. [22]). From Cameron's operator Ax = z in (5.1),
EE(-1)n-fcXn-fc Zfc = 1 .
n=0k=0
Hence, for n > 1
]T(-1)n-kXn-kZk =0 .
k=0
When x0 = z0 = 1, we have the following inversion formula.
0
a
n
106
Ars Math. Contemp. 18 (2020) 105-115
Lemma 5.2.
If Xn
'.. 1
Z2 Z1
then zn
X1 X2
.. 1
X2 X1
From Trudi's formula, it is possible to give the combinatorial expression
E
Î1+2Î2+-----hnt„=n
tl +	+ tn \ (_l)n —ti-----tnjlj2... „tn
tl , . . . , tn
z1 z2 •••zn
By applying these lemmas to Theorem 4.1, we obtain an explicit expression for the hyper-geometric degenerate Bernoulli numbers.
Theorem 5.3. For N, n > 1,
ßN,n(A) = n! £ (V"^)
, , , , .	\ 117 ... 7tn /
t 1 +2t2+-----+ntn=n v	7
(_1)
t i+---+t„
(1 _ NA)N!\t1 /(1 _ NA|A)2N!\t2 /(1 _ NA|A) (N + 1)! J V (N + 2)! )	(N + n)!
*N !
Theorem 5.4. For N, n > 1,
(_1)"(1 _ NA|A)WN! (N + n)!
ßN,l(A) 1 ßN#) ßN,i(A)
ßN,n-l(A) ß»,n-2(A)
(n-1)! ßN,n(A) n!
(n —2)! ß N ,n — l(A) (n—1)!
1 0 ßN,1(A) 1 ßN#1 ßN,1(A)
Applying the Trudi's formula in Lemma 5.1 to Theorem 5.4, we get the inversion relation of Theorem 5.3.
Theorem 5.5. For N, n > 1,
(1 _ NA|A)nN! (N + n)!
E
ti+2t2+-----+nt„=n
t1 + • • • + tr t1 , . . . , tn
(_1)
ti + --- + t„
X (ßN,1(A))
t1 f ßN,2 (AU t2 /ßN,n(AU tn
2!
n!
1
1
z
X
n
n
xn

X
0
T. Komatsu: Hypergeometric degenerate Bernoulli polynomials and numbers	173
6 Coefficients of hypergeometric degenerate Bernoulli numbers
Hypergeometric Cauchy polynomials cN,n(x) ([ ]) have similar properties. The generating function is given by
1
(1+ t)x2Fi(1,N; N + 1; -t)
TO
E
n=0
tn
CN,n (x)—7 . n!
(6.1)
When x = 0 in (6.1), cN,n = cN,n(0) are the hypergeometric Cauchy numbers ([ , 13, 14, 15, 19]). When N = 1 in (6.1), cn(x) — ci,n(x) are the ordinary Cauchy polynomials (e.g., [ ]). When x = 0 and N =1 in (6.1), cn = c1jK(0) are the classical Cauchy numbers (see, e.g., [10, Chapter VII]), defined by
t
_ t„
Ec„ .
n!
log(1+ t)
(6.2)
n=0
The number cn/n! is sometimes referred to as the Bernoulli number of the second kind (see, e.g. [16]). A determinant expression of hypergeometric Cauchy numbers ([1, 23]) is given by
" 1

N
N+1
N+2
0
N
N+n-N
N N+1
N
N+n-2 N-
N+n
1
N
NN1
N+2
0
1
N N +1
ci,
(6.3)
was discovered by
N+n-1
The determinant expression for the classical Cauchy numbers cn Glaisher ([11, p. 50]). A more general case is considered in [21].
From the expression in Theorem 5.3, the hypergeometric degenerate Bernoulli number ^N,n is a polynomial in A with rational coefficients and degree at most n. Thus, we can write
£N,n(A) = d„,„A" + dn,n-lA"-1 + • • • + d„,iA + d„,0 .	(6.4)
In this section, we give some coefficients explicitly. By this theorem, we can see that hypergeometric degenerate Bernoulli numbers are closely related with both hypergeometric Bernoulli numbers and hypergeometric Cauchy numbers.
Theorem 6.1. For N > 1 and n > 0,we have
d„,n = CN,n and d„,0 = B
>N,n
n!
1
Remark 6.2. When N =1, Theorem 6.1 is reduced to [ , Theorem 3.1]. This implies that the leading coefficient of ^„(A) is equal to the n-th Cauchy number cn and the constant term is equal to the n-th Bernoulli number B„.
Proof of Theorem 6.1. Since
%—k
(1 - NA|A)„-k = An-k^(-1)
k—i
l=0
-k r i i l n — k
E
l=0
(-1)
n — k l
n-k-i
i=0
A - n
\n-k-^N l-i
i
106
Ars Math. Contemp. 18 (2020) 105-115
by Proposition 2.1 we obtain for n > 1
n—1
(1- NAJA)I—ff! ^(a)
k=0 n- 1
-E
(N + n - k)!k! N (A)
(6.5)
— k r	,n l
n — k
(N + n - k)!k!
k=0 v 7 l=0
E ' -k E(-1)
\n—k—i
i=0
^n—k—¿n l—i
Note that ^w,o(A) = 1.
For constant term of the polynomial in A, as i = n — k in (6.5)
n— 1
d0,0	(1 - NA|A)n—kN!
~nr = - Twrr.—^rdk,0(A)
k=0 n1
(N + n - k)!k!
k
EN !dk,0 fc_Q (N + n - k)!k! ^
n - k l
l
n — k
N
l—n+k
n— 1
-
k=0
N !d
k,0
(N + n - k)!k! '
Hence,
E
k=0
N + n k
dk,0 = 0
with d0 0 = 1. Since the hypergeometric Bernoulli numbers BN,n satisfies the same recurrence relation, namely,
E
k=0
N + n k
BN,k = 0
with BN 0 = 1 ([ , Proposition 1], [18, (6)]), we can conclude that
dn,0 = Bn,; .
For the leading coefficient, that is, the coefficient of An of the polynomial in A, as i = 0 in (6.5)
d;
1
-
k=0
(-1)n—k N !d, (N + n - k)!k!
—k
k,k
l=0
n - k l
Nl
n— (-1)n—k n !(N )(n—k) - / y —-rrm— dk,k
k=0 n—1
(N + n - k)!k!
(-1)n—kN .
- 2-^1 TaTVZ.-LYÛdk,k .
Thus, for n > 1
k=0
n
(N + n - k)k!
y (-1)n—kN d =0
Z^(N + n - k)k! dk,k =0
n!
l
n!
T. Komatsu: Hypergeometric degenerate Bernoulli polynomials and numbers	175
or
E
k=0
(-1)k
(N + n - k)k!
dk,k — 0
with do,o = 1. Since the hypergeometric Cauchy numbers cN,n satisfies the same recurrence relation, namely,
(-1)k
E
k=0
(N + n - k)k!
CN,k — 0
with cN,0 — 1 ([ , Proposition 1]), we can conclude that
dn,n cN,n •
□
6.1 Another method
Howard [16] found explicit formulas for all the coefficients by proving the following. For
n > 2
n1
^n(A)— CnAn + £(-1)n-j-Bj j = 1 j
j - 1
An-j
(6.6)
where are the Stirling numbers of the first kind, determined by
c(x - 1) • • • (x - n + 1) — E(-1)
n-k
k=0
n k
xk .
(6.7)
and cn are the Cauchy numbers (of the firs kind), defined by the generating function
log(1+1)
ECn .
n!
(6.8)
We have another expression of the coefficients of ^N,n(A) directly from Theorem 2.2. Since by (6.7)
(1 - NA|A)i — E(-1)i-j E
j=0
1=1
E(-1)i-j E
j=0 l=j
Nl-j • Ai-j
Nl-j • Ai-j .
we have for j — 0,1,..., n
tJn,n-j
!E
k= 1
(-N !)k(-1)n-j
jT
E
¿1+—+ik=n i1,...,ik>i
(N + ii)! ••• (N + ik )!
ii
x dxj US
E
Vifc=1
x=N
(6.9)
t
j
1

x
x
106
Ars Math. Contemp. 18 (2020) 105-115
If j = n in (6.9), we get the coefficient of the constant in A as
n	.
n!
dn,0 = n!^(-N !)k ]T
k	- + +. n!(N + n)! ••• (N + ifc)!'
k= 1 il +-----+ik = n ^ ' y '
i1,...,ik>i
which is equal to BNn by (2.4). If j = 0 in (6.9), we get the leading coefficient in A as
A	^ (N)(i 1) (N)(ifc)
d,„ = n- 1= (-N!)k(-1)n E (NTH)i-(U^
i1,...,ifc>1
n Nk
= n!V(-:L)n-k V _-_
! k==1( ) -	(N + ii) • • • (N + ik) '
k = 1 -----=n
i1,...,ik>1
which is equal to cN,n in [1, 23].
However, it seems difficult to express other terms of ^N,n(A) in any explicit form,
except the leading coefficient and the constant.
ORCID iDs
Takao Komatsu © https://orcid.org/0000-0001-6204-5368 References
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