Generalizations of poly-Bernoulli and poly-Cauchy numbers

Abstract

In this paper we consider some generalizations of poly-Bernoulli and poly-Cauchy numbers. The first is by means of the Hurwitz–Lerch zeta function. The second generalization is via weighted Stirling numbers. The third one is given with the help of degenerate Stirling numbers. All these generalizations lead to symmetries between various types of Stirling numbers, and enable us to investigate and expand algebraic properties of poly-Bernoulli and poly-Cauchy numbers. We also combine these generalizations and derive numerous combinatorial and arithmetical identities.

1 Introduction

The Bernoulli numbers, $B_{n}$, $n\in {\mathbb {Z}}$, $n\geqslant 0$, defined by the generating function

play important roles in many applications in number theory, combinatorics, numerical analysis and other branches of mathematics. The first few Bernoulli numbers are $B_{0}=1$, , $B_{2}= 1/6$, and $B_{n}=0$ for all odd $n\geqslant 3$.

Bernoulli numbers have several relations with combinatorial numbers. For example, we have

(1)

where $S_{2}( n,m)$ denotes the Stirling numbers of the second kind which are defined as

(2)

These numbers are also determined by

and $S_{2}( n,m) =0$ for $n<m$ (see [13] for other relations and properties).

There are numerous generalizations of Bernoulli numbers, including the poly-Bernoulli numbers defined by Kaneko [19] in 1997 as

where

$$\begin{aligned} {{\mathrm {Li}}}_{k}( z) =\sum \limits _{n=1}^{\infty }\frac{z^{n}}{n^{k} } \end{aligned}$$

is the kth polylogarithm function. When $k=1$ we have $B_{n}^{( 1) }=B_{n}$, the classical Bernoulli number, except that when $n=1$. The generating function of the poly-Bernoulli numbers may also be written in terms of iterated integrals as

$$\begin{aligned} {\mathrm {e}}^{t}\frac{1}{{\mathrm {e}}^{t}-1}\,\underset{( k-1) \text {-times}}{\underbrace{\int \limits _{0}^{t}\frac{1}{{\mathrm {e}}^{u}-1} \int \limits _{0}^{t} \frac{1}{{\mathrm {e}}^{u}-1}\cdots \int \limits _{0}^{t}\frac{ u}{{\mathrm {e}}^{u}-1}}}\underset{( k-1) \text {-times}}{\underbrace{ \,{\mathrm {d}}u{\mathrm {d}}u\cdots {\mathrm {d}}u}}=\sum \limits _{n=0}^{\infty }B_{n}^{( k) }\frac{t^{n}}{n!}, \end{aligned}$$

and a combinatorial formula for $B_{n}^{( k) }$ is

(3)

Thus putting $k=1$ gives (1). The poly-Bernoulli numbers are related to multiple zeta values (see [2, 3]), and for comprehensive information we refer to [1, Chapter 14].

The Cauchy numbers are denoted by $c_{n}$ and defined as the definite integral of the falling factorial

$$\begin{aligned} c_{n}=\int \limits _{0}^{1}t( t-1) \cdots ( t-n+1)\, {\mathrm {d}}t=n!\int \limits _{0}^{1}\left( {\begin{array}{c}t\\ n\end{array}}\right) \,{\mathrm {d}}t \end{aligned}$$

(see [27]). The generating function of Cauchy numbers is given by

Cauchy numbers share a particular relationship with Bernoulli numbers of the second kind $b_{n}$, in that (for detailed information on $b_{n}$ we refer to [6, 17, 28]). Cauchy numbers satisfy the combinatorial formula

where $S_{1}( n,m)$ are the (unsigned) Stirling numbers of the first kind which are defined by

These numbers also arise as the coefficients of the rising factorial

and $S_{1}( n,m) =0$ if $n<m$.

The definite integral defining the Cauchy numbers and the iterated integral expression for poly-Bernoulli numbers motivate a definition of poly-Cauchy numbers as

(see [22]). For $k=1$, $c_{n}^{( 1) }=c_{n}$ are the Cauchy numbers. The poly-Cauchy numbers and the Stirling numbers of the first kind share a particular relationship in that

(4)

To give a generating function for poly-Cauchy numbers, let ${\mathrm {Lif}}_{k}( z)$ be the polylogarithm factorial function defined by

Then we have

Recently there have been extensive studies in generalizing poly-Bernoulli and poly-Cauchy numbers. In [9], poly-Bernoulli numbers are generalized with a q parameter. In [25], shifted poly-Cauchy and poly-Bernoulli numbers are defined and in [23] these numbers are further generalized with a q parameter. In [12, 16], poly-Bernoulli and poly-Cauchy numbers and polynomials are considered by means of multiparameters.

The objective of this paper is to give further generalizations for poly-Bernoulli and poly-Cauchy numbers. These generalizations, in particular, give some symmetries for Stirling number series, and lead to a unified investigation of arithmetic and algebraic properties for poly-Bernoulli and poly-Cauchy numbers. Moreover since the multiple zeta values and the Arakawa–Kaneko zeta functions are closely related to poly-Bernoulli numbers and polynomials, inverse binomial series and Bernoulli polynomial series (see [3, 10, 11, 32, 33]), the generalizations in this paper may lead to further investigations related to various zeta functions. One of the generalizations is inspired from the polylogarithm function. Since the polylogarithm function can be derived from the Hurwitz–Lerch zeta function, poly-Bernoulli numbers are generalized by using this more general function. Similar modification leads to a generalization for poly-Cauchy numbers. The other generalizations come from the combinatorial identities (3) and (4). These formulas, which involve Stirling numbers, are reconsidered by employing weighted and degenerate weighted Stirling numbers. Numerous formulas such as a recurrence formula and a symmetry for generalizations of Stirling numbers, and some divisibility properties including partial congruences are also presented.

2 Generalizations with Hurwitz–Lerch zeta function

A general Hurwitz–Lerch zeta function ${\Phi } ( z,s,a)$ is defined for $s\in {\mathbb {C}}$ when $\vert z\vert <1$, ${\text {Re}}\, s >1$ when $\vert z\vert =1$ and by the series ([30])

This function includes several special functions, one of which is the polylogarithm function

(5)

Since

equation (5) motivates the generalization

(6)

so that $B_{n}^{( k) }( 1) =B_{n}^{( k) }$. We call $B_{n}^{( k) }( a)$ the Hurwitz type poly-Bernoulli numbers. An explicit formula can be obtained by direct use of (2) and (6) as follows

Theorem 2.1

For the Hurwitz type poly-Bernoulli numbers $B_{n}^{( k) }( a)$ we have

(7)

Proof

We have

by means of standard operation on series (see, for instance [20, Equation (1.11)]). Comparison of the coefficients of in the first and last terms in this string of equalities yields (7). $\square$

One of the remarkable properties of poly-Bernoulli numbers with combinatorial interpretations is a closed formula in the case of negative upper index k (for combinatorial interpretations of poly-Bernoulli numbers we refer to [5, 26]). Such a formula for Hurwitz type poly-Bernoulli numbers may be given in terms of weighted Stirling numbers. Carlitz [8] has defined $R_{1}( n,k,x)$ and R( n, k, x) by means of

and

For convenience we call $R_{1}( n,k,x)$ and R( n, k, x) the weighted Stirling numbers of the first and second kind and denote them by $S_{1}( n,k,x)$ and $S_{2}( n,k,x)$, respectively, although Carlitz used that terminology for slightly different numbers. When $x=0$ we have $S_{1}( n,k,0) =( -1) ^{n-k}S_{1}( n,k)$ and $S_{2}( n,k,0) =S_{2}( n,k)$, the ordinary Stirling numbers.

Theorem 2.2

For the Hurwitz type poly-Bernoulli numbers $B_{n}^{(-k) }( a)$ we have

(8)

For the proof we need two relations regarding Stirling and weighted Stirling numbers of the second kind.

Lemma 2.3

We have

(9)

and

(10)

Proof

Although the first equality (9) can be found in standard texts such as [13] we give the proof here in order to show the relation to the second equation. We start with (2). Replacing m with $j+1$,

Differentiation with respect to t gives

Now,

Comparing the coefficients of on both sides gives the first part of (9). The second part follows from

in a somewhat more direct manner than (9). We have

Comparison of the coefficients gives the result. $\square$

Proof of Theorem 2.2

In order to prove the theorem we calculate the generating function of $B_{n}^{( -k) }( a)$ in the following form:

From (9) and (10) we get

Comparing the coefficients gives the result. $\square$

We note that for $a=1$, Theorem 2.2 reduces to the closed formula given by Arakawa and Kaneko [2] and by Sánchez-Peregrino [29].

Recall that the poly-Cauchy numbers $c_{n}^{( k) }$ are defined by means of the generating function

where ${\mathrm {Lif}}_{k}( z)$ is the polylogarithm factorial function given by

In order to give a Hurwitz type extension of poly-Cauchy numbers we define the Hurwitz–Lerch factorial zeta function by

For $a=1$ and $s=k$, a natural number, we obviously have . Thus the Hurwitz type poly-Cauchy numbers which are denoted by $c_{n}^{( k) }( a)$ may be defined by

As in the Bernoulli case an explicit formula for the Hurwitz type poly-Cauchy numbers can be obtained involving Stirling numbers of the first kind.

Theorem 2.4

For the Hurwitz type poly-Cauchy numbers $c_{n}^{(k) }( a)$ we have

(11)

Proof

We have

Equating the coefficients gives (11). $\square$

We remark that the Hurwitz type poly-Cauchy numbers are defined as shifted poly-Cauchy numbers in [25] and a generalization is given in [23].

Komatsu [22] defined the poly-Cauchy numbers of the second kind $\widehat{c}_{n}^{\,( k) }$ by

Following Komatsu we define the Hurwitz type poly-Cauchy numbers of the second kind as

An explicit formula for $\widehat{c}_{n}^{\,( k) }( a)$ is

(12)

which can be shown along the lines of the proof of Theorem 2.4.

Orthogonality and inverse relations for Stirling numbers allow us to present some relations for Hurwitz type poly-Bernoulli and poly-Cauchy numbers. From the orthogonality relations ([13, p. 264])

$\delta _{mn}$ being the Kronecker symbol (that is, $\delta _{mn}=1$ for $m=n$ and $\delta _{mn}=0$ otherwise), inverse relations

(13)

follow directly. Using (13) we get the following results.

Theorem 2.5

For the Hurwitz type poly-Bernoulli numbers $B_{n}^{( k) }( a)$ we have

(14)

Proof

This follows from (13) by taking $f_{m}=( -1) ^{m}m!/( m+a) ^{k}$ and $g_{n}=( -1) ^{n}B_{n}^{( k) }( a)$. $\square$

Theorem 2.6

For the Hurwitz type poly-Cauchy numbers we have

(15)

and

(16)

Proof

From the definitions, (15) follows from the duality relation (13) by taking $g_{m}=1/( m+a) ^{k}$ and $f_{n}=c_{n}^{( k) }( a)$, and (16) by taking $g_{m}=( -1) ^{m}/( m+a) ^{k}$ and $f_{n}=\widehat{c}_{n}^{\,( k) }( a)$.$\square$

Theorem 2.7

For nonnegative integer n we have

$$\begin{aligned} B_{n}^{( k) }( a)= & {} \sum _{l=0}^{n}\sum _{m=0}^{n}( -1) ^{m+n}m!S_{2}( n,m) S_{2}( m,l) c_{l}^{( k) }( a),\end{aligned}$$

(17)

$$\begin{aligned} B_{n}^{( k) }( a)= & {} \sum _{l=0}^{n}\sum _{m=0}^{n}( -1) ^{m}m!S_{2}( n,m) S_{2}( m,l) \widehat{c}_{l}^{\,( k) }( a),\end{aligned}$$

(18)

$$\begin{aligned} c_{n}^{( k) }( a)= & {} \sum _{l=0}^{n}\sum _{m=0}^{n}\frac{( -1) ^{m+n}}{m!}\,S_{1}( n,m) S_{1}( m,l) B_{l}^{( k) }( a),\end{aligned}$$

(19)

$$\begin{aligned} \widehat{c}_{n}^{\,( k) }( a)= & {} \sum _{l=0}^{n}\sum _{m=0}^{n}\frac{( -1) ^{n}}{m!}\,S_{1}( n,m) S_{1}( m,l) B_{l}^{( k) }( a). \end{aligned}$$

(20)

Proof

Using (15) we compute

giving (17). Similarly, use of (16) gives (18). On the other hand, by (14) we get (19) and (20). $\square$

There are no closed formulas for the poly-Cauchy numbers like those for the poly-Bernoulli numbers in (8). Instead we have

and

3 Generalizations with weighted Stirling numbers

In this section we consider generalizations of poly-Bernoulli and poly-Cauchy numbers motivated by (3) and (4). We first review some facts about weighted Stirling numbers. As mentioned in the previous section the weighted Stirling numbers of the second kind $S_{2}( n,m,x)$ are defined by

Thus the weighted Stirling numbers of the second kind are in fact ordinary Stirling numbers of the second kind with an exponential factor. So the weighted and ordinary Stirling numbers of the second kind are related by

(21)

From (21) we have $S_{2}( n,m,x) =0$ for $n<m$. Since

we have

(22)

Multiplying (21) by $m!\left( {\begin{array}{c}y\\ m\end{array}}\right)$ and summing over m gives

(23)

Equation (23) gives the recurrence formula

(24)

From (22) or (24) with $S_{2}( 0,0,x) =1$ we have

For the weighted Stirling numbers of the first kind $S_{1}( n,m,x)$ defined in Sect. 2 by

we have

(25)

where $\langle x\rangle _{r}$ is the rising factorial

$$\begin{aligned} \langle x\rangle _{r}=x( x+1) \cdots ( x+r-1) \end{aligned}$$

with $\langle x\rangle _{0}=1$,

(26)

and

$$\begin{aligned} S_{1}( n,0,x) =\langle x\rangle _{n},\qquad S_{1}( n,n,x) =1. \end{aligned}$$

We note from (25) that $S_{1}( n,m,x) =0$ for $n<m$.

Recall from (3) that

In this equation we replace $S_{2}( n,m)$ by weighted Stirling number of the second kind $S_{2}( n,m,x)$. We then define weighed type poly-Bernoulli numbers by

(27)

From (27) we have

We remark that this kind of generalization for poly-Bernoulli numbers has recently been studied by Coppo and Candelpergher [10], Bayad and Hamahata [4], and Komatsu and Luca [24]. They called $B_{n}^{( k) }( x)$ the poly-Bernoulli polynomials. To be more precise Coppo and Candelpergher defined these polynomials as

while Bayad and Hamahata defined them by

and Komatsu and Luca defined them by

The numbers $B_{n}^{( -k) }( x)$ satisfy a closed formula like (8).

Theorem 3.1

For nonnegative integers n and k we have

Proof

We have

since $S_{2}( n,m,x) =0$ for $n<m$ and $S_{2}( k,j,1) =S_{2}( k+1,j+1)$. The result now follows by comparing the coefficients. $\square$

Recall from (4) that

This last equation was given by Komatsu [22] and can be derived from (12) with $a=1$. Writing $S_{1}( n,m,x)$ instead of $S_{1}( n,m)$, we define weighted type poly-Cauchy numbers of the first kind and of the second kind by

(28)

and

From (28) and (26) we have

We also have

In the view of above equations we note that Komatsu [21, 22], Kamano and Komatsu [18], and Komatsu and Luca [24] have given these definitions and called them poly-Cauchy polynomials.

For the weighted type poly-Cauchy numbers of the second kind we similarly have

and

In order to obtain combinatorial formulas for the weighted type poly-Bernoulli and poly-Cauchy numbers, we use the orthogonality and inverse relations for weighted Stirling numbers.

Orthogonality relations for weighted Stirling numbers have been given by Carlitz [8] as

We thus obtain the inverse relation

(29)

Theorem 3.2

For the weighted type poly-Bernoulli numbers $B_{n}^{( k) }( x)$ we have

Proof

This follows from relation (29) by taking $f_{m}=( -1) ^{m}m!/( m+1) ^{k}$ and $g_{n}=( -1) ^{n}B_{n}^{( k) }( x)$. $\square$

Theorem 3.3

For the weighted type poly-Cauchy numbers of the first and second kind we have

(30)

and

(31)

Proof

From the definitions, (30) follows from (29) by taking $g_{m}=1/( m+1) ^{k}$ and $f_{n}=c_{n}^{( k) }( x)$, and (31) by taking $g_{m}=( -1) ^{m}/( m+1)^{k}$ and $f_{n}=\widehat{c}_{n}^{\,( k) }( x)$. $\square$

Theorem 3.4

For nonnegative integer n and any x, y we have

Proof

The definitions of $B_{n}^{( k) }( x) , c_{n}^{\,( k) }( x)$ and $\widehat{c}_{n}^{\,( k) }( x)$, together with Theorems 3.2 and 3.3 give the results. $\square$

4 Hurwitz type weighted poly-Bernoulli and poly-Cauchy numbers

As the title indicates, we may combine the definitions of Hurwitz type and weighted type poly-Bernoulli and poly-Cauchy numbers. We define

From the generating functions of Hurwitz type and weighted type numbers we obtain

The generating functions of weighted Stirling numbers suggest two different recurrence formulas, namely,

and

These recurrences lead to the following interesting results:

5 Degenerate poly-Bernoulli and poly-Cauchy numbers

The method that extends Bernoulli and Cauchy numbers to poly-Bernoulli and poly-Cauchy numbers can be adapted to different kinds of number sequences. In this section we consider poly-extensions of degenerate Bernoulli and Cauchy numbers.

The degenerate Bernoulli numbers $\beta _{n}( \lambda )$ were defined by Carlitz [7] as

$$\begin{aligned} \frac{t}{( 1+\lambda t) ^{\mu }-1}=\sum _{n=0}^{\infty }\beta _{n}( \lambda )\, \frac{t^{n}}{n!} \end{aligned}$$

where $\lambda \mu =1$. Note that the limiting case $\lambda =0$ gives the generating function of the ordinary Bernoulli numbers, so $\beta _{n}( 0) =B_{n}$.

Following Kaneko we define the degenerate poly-Bernoulli numbers $\beta _{n}^{( k) }( \lambda )$ by

(32)

Carlitz [7] also defined degenerate Stirling numbers of the second kind by

(33)

Equations (32) and (33) give an explicit formula for degenerate poly-Bernoulli numbers.

Theorem 5.1

For the degenerate poly-Bernoulli numbers $\beta _{n}^{(k) }( \lambda )$ we have

(34)

Proof

We have

Comparing the coefficients gives the result. $\square$

We recall that the poly-Cauchy numbers are defined by

We therefore define degenerate poly-Cauchy numbers $c_{n}^{( k) }( \lambda )$ by

To obtain an explicit formula for $c_{n}^{( k) }( \lambda )$ we use the degenerate Stirling numbers of the first kind which were defined by Carlitz [7] as

Theorem 5.2

For the degenerate poly-Cauchy numbers $c_{n}^{( k) }( \lambda )$ we have

(35)

Proof

We have

Comparing the coefficients proves the theorem. $\square$

The degenerate Stirling numbers of the first and second kind satisfy the orthogonality relation

(36)

([7, Equation (1.14)]). From (36) we obtain the inverse relation

(37)

which provides the following results.

Theorem 5.3

For degenerate poly-Bernoulli and poly-Cauchy numbers we have

(38)

and

(39)

Proof

Relation (38) follows from (37) with $f_{m}=( -1) ^{m}m!/( m+1) ^{k}$ and $g_{n}=( -1) ^{n}\beta _{n}^{( k) }( \lambda )$, while relation (39) follows from (37) with $g_{m}=1/( m+1) ^{k}$ and $f_{n}=c_{n}^{( k) }( \lambda )$. $\square$

Theorem 5.4

For nonnegative integer n and any $\lambda , \theta$ we have

(40)

(41)

Proof

For (41) we use (39) and obtain

Similarly (38) gives (40). $\square$

6 Degenerate weighted poly-Bernoulli and poly-Cauchy numbers

We may further generalize poly-Bernoulli and poly-Cauchy numbers by means of degenerate weighted Stirling numbers. The degenerate weighted Stirling numbers of the first and second kind are defined by Howard [14] respectively as

(42)

and

(43)

We define degenerate weighted poly-Bernoulli $\beta _{n}^{( k) }( x,\lambda )$ and poly-Cauchy numbers $c_{n}^{( k) }( x,\lambda )$ by simply replacing the degenerate Stirling numbers in (34) and (35) with and . Thus

(44)

are the degenerate weighted poly-Bernoulli numbers and

(45)

are the degenerate weighted poly-Cauchy numbers. Generating functions for these numbers are given as follows

Theorem 6.1

We have

and

Proof

From (43) and (44) we have

Similarly (42) and (45) yield the second result. $\square$

From Theorem 6.1 we observe that $\beta _{n}^{( k) }( 0,\lambda ) =\beta _{n}^{( k) }( \lambda )$ and $c_{n}^{( k) }( \lambda ,\lambda ) =c_{n}^{( k) }( \lambda )$.

In order to obtain relations among degenerate weighted poly-Bernoulli and poly-Cauchy numbers, we need an inverse relation for the degenerate weighted Stirling numbers. Hsu and Shiue [15] defined a very general class $S( n,m;\alpha ,\beta ,x)$ of sequences which generalize the Stirling numbers. An exponential generating function for $S( n,m;\alpha ,\beta ,x)$ is

(46)

We note from (46) that and . They also gave the orthogonality relation

(47)

If we write $\alpha =1$, $\beta =\lambda$ and replace x by in (47) we obtain the orthogonality relation for the degenerate Stirling numbers as

(48)

From (48) we easily derive the inverse relation

(49)

Theorem 6.2

For degenerate weighted poly-Bernoulli and poly-Cauchy numbers we have

(50)

and

(51)

Proof

Relation (50) follows from (49) by setting and $g_{n}=( -1) ^{n}\beta _{n}^{( k) }( x,\lambda )$. Relation (51) is obtained from (49) by taking $g_{m}=1/( m+1) ^{k}$ and $f_{n}=c_{n}^{( k) }( x,\lambda )$. $\square$

Theorem 6.3

For nonnegative integer n and any values of $x,y,\lambda ,\theta$ we have

Proof

Using Theorem 6.2,

giving the first part. Similarly,

gives the second part. $\square$

7 Congruences

The presence of Stirling numbers in explicit formulas for weighted, degenerate and degenerate weighted poly-Bernoulli and poly-Cauchy numbers enables us to discuss some divisibility properties for them. These properties give information about the denominators of these numbers. The denominators of classical Bernoulli numbers are completely determined by the von Staudt and Clausen theorem, which not only determines the denominator but also describes the fractional part of $B_{n}$. In [19] the denominators of di-Bernoulli numbers $B_{n}^{( 2) }$ are completely determined while in [2] partial results for denominators of poly-Bernoulli numbers are obtained.

In this section p will denote a prime number, ${\mathbb {Z}}_{p}$ the ring of p -adic integers and ${\mathbb {Z}}_{p}^{\times }$ the multiplicative group of units in ${\mathbb {Z}}_{p}$. The p-adic valuation “${\mathrm { ord}}_{p}$” is defined by setting ${\mathrm {ord}}_{p}( x) =k$ if $x=p^{k}y$ with $y\in {\mathbb {Z}}_{p}^{\times }$ and ${\mathrm {ord}}_{p}( 0 ) =\infty$. A congruence $x\equiv y\,(\mathrm{mod}\ {m{\mathbb {Z}}_{p}})$ is equivalent to ${\mathrm {ord}}_{p}( x-y) \geqslant {\mathrm {ord}}_{p}( m)$, and if x and y are rational numbers this congruence for all primes p is equivalent to the congruence $x\equiv y\,(\mathrm{mod}\ {m})$ given by

$$\begin{aligned} \frac{a}{b}\equiv \frac{c}{d}(\mathrm{mod }m)\qquad \Longleftrightarrow \qquad m\mid ( ad-bc), \end{aligned}$$

whenever p does not divide b and d.

Since degenerate weighted poly-Bernoulli and poly-Cauchy numbers are the most general cases considered so far, we present congruences for them with proofs and state the results for others as corollaries.

Theorem 7.1

Let p be an odd prime number. For any p-adic integer x and $\lambda \in {\mathbb {Z}}_{p}$ we have

$$\begin{aligned} p^{k-1}\beta _{p}^{( k) }( x,\lambda ) \equiv p^{k-1}\biggl ( \frac{1}{2^{k}}-x\biggr ) -\frac{[ ( p-1) ( 1-\lambda ) +2x] ( p-1) !}{2} (\mathrm{mod }p^{k}{\mathbb {Z}}_{p}) \end{aligned}$$

if $\lambda \in p{\mathbb {Z}}_{p}$ and

$$\begin{aligned} p^{k-1}\beta _{p}^{( k) }( x,\lambda ) \equiv -\,\frac{[ ( p-1) ( 1-\lambda ) +2x] ( p-1) !}{2}(\mathrm{mod }p^{k}{\mathbb {Z}}_{p}) \end{aligned}$$

if $\lambda \in {\mathbb {Z}}_{p}^{\times }$.

Proof

We write (44) as

Since x is a p-adic integer, for $1<m<p$ ([15, Theorem 3], [31, Equation (3.4)]). We also have ${\mathrm {ord}}_{p}( m!) =0$ and ${\mathrm {ord}}_{p}( ( m+1) ^{k}) =0$ for $1<m<p-1$. Thus the latter sum is in $p{\mathbb {Z}}_{p}$. For the terms and we use

([14, Equation (4.2)]) and obtain

From (43) we easily get , and repetitive application of the recurrence formula

([14, Equation(4.11)]) gives

Thus we have

since $p!\in p{\mathbb {Z}}_{p}$. Now, since

we have

if $\lambda \in p{\mathbb {Z}}_{p}$ and

if $\lambda \in {\mathbb {Z}}_{p}^{\times }$. $\square$

For $x=0$ we have the following result for degenerate poly-Bernoulli numbers.

Corollary 7.2

For an odd prime p and $\lambda \in {\mathbb {Z}}_{p}$ we have

if $\lambda \in p{\mathbb {Z}}_{p}$ and

if $\lambda \in {\mathbb {Z}}_{p}^{\times }$.

Similarly, if we let $\lambda =0$ in Theorem 7.1 we obtain

Corollary 7.3

For an odd prime p and any p-adic integer x we have

We note that similar congruences for poly-Bernoulli numbers were given in [2] and [19].

The following result gives a divisibility property for degenerate weighted poly-Cauchy numbers of the first kind.

Theorem 7.4

Let p be an odd prime number. For any p-adic integer x and $\lambda \in {\mathbb {Z}}_{p}^{\times }$ we have

Proof

Consider the defining equation

Since $\lambda \in {\mathbb {Z}}_{p}^{\times }$ and $\lambda -x\in {\mathbb {Z}} _{p}$ we have for $1<m<p$ ([15, Theorem 3], [31, Equation (3.4)]). Thus we get

From the equation

([14, Equation(4.1)]) we obtain

From (42) we have and

([14, p. 51]) gives

since . Thus we have

Now, since for any $z\in {\mathbb {Z}}_{p}$ we have $( z) _{p}\in p {\mathbb {Z}}_{p}$, we see that $p^{k-1}( x-\lambda ) _{p}$ and $p^{k-1}( 2\lambda -x) _{p}$ are zero modulo $p^{k}$. Therefore we obtain

which is the result. $\square$

For $\lambda =0$ we have the following result.

Corollary 7.5

For an odd prime p and any p-adic integer x we have

The next result states a divisibility property for $c_{n}^{( k) }( \lambda )$, which follows from Theorem 7.4 with $x=\lambda$.

Corollary 7.6

Let p be an odd prime. Then

if $\lambda \in {\mathbb {Z}}_{p}^{\times }$ and

if $\lambda \in p{\mathbb {Z}}_{p}$.

For the second result we note that if $\lambda \in p{\mathbb {Z}}_{p}$ then is in ${\mathbb {Z}}_{p}$ not in $p{\mathbb {Z}} _{p}$. Thus when $\lambda \in p{\mathbb {Z}}_{p}$ we conclude that