Recurrences and Congruences for Higher order Geometric Polynomials and Related Numbers

We obtain new recurrence relations, an explicit formula, and convolution identities for higher order geometric polynomials. These relations generalize known results for geometric polynomials, and lead to congruences for higher order geometric polynomials, particularly for p-Bernoulli numbers.


Introduction
For a complex variable y, the geometric polynomials w n (y) of degree n are defined by [31] w n (y) = n k=0 n k k!y k , (1.1) where n k is the Stirling number of the second kind [15]. These polynomials have been studied from analytic, combinatoric, and number theoretic points of view. Analytically, they are used in evaluating geometric series of the form [4] ∞ k=0 k n y k , with y d dy k n y k = 1 1 − y w n y 1 − y .
for every |y| < 1 and every n ∈ Z, n ≥ 0. Combinatorially, they are related to the total number of preferential arrangements of n objects w n (1) := w n = n k=0 n k k!, that is, the number of partitions of an n-element set into k nonempty distinguishable subsets (c.f. [10]). Number theoretic studies on the geometric polynomials are mostly originated from their exponential generating function ∞ n=0 w n (y) t n n! = 1 1 − y (e t − 1) .
For example, setting y = − 1 2 gives where B n are Bernoulli numbers and T n are tangent numbers. Bernoulli numbers also occur in integrals involving geometric polynomials, namely we have [24] 1 0 w n (−y) dy = B n , n > 0.
Moreover, we note that [21] 1 0 (1 − y) p w n (−y) dy = 1 p + 1 B n,p , where B n,p are p-Bernoulli numbers [30] (see Section 2 for definitions). The congruence identities of geometric numbers is also one of the subjects studied. Gross [16] showed that w n+4 = w n (mod 10) , which was generalized by Kauffman [19] later. Mező [27] also gave an elementary proof for Gross' identity. Moreover, Diagana and Maïga [11] used p-adic Laplace transform and p-adic integration to give some congruences for geometric numbers. We refer the papers [5,6,7,12,20,29] and the references therein more on geometric numbers and polynomials. In the literature, there are numerous studies for the generalization of geometric polynomials (e.g. [13,14,22,23]). One of the natural extension of geometric polynomials is the higher order geometric polynomials [4] w (r) n (y) = n k=0 n k (r) k y k , r > 0, (1.2) where (x) n is the Pochhammer symbol defined by (x) n = x (x + 1) · · · (x + n − 1) with (x) 0 = 1. It is evident that w (1) n (y) = w n (y). The polynomials w (r) n (y) have the property y d dy for any n, r = 0, 1, 2, . . ., and may be defined by means of the exponential generating function ( On the other hand, the higher order geometric polynomials and exponential (or single variable Bell) polynomials ϕ n (y) = n k=0 n k y k are connected by (c.f. [4]). According to this integral representation, several generating functions and recurrence relations for higher order geometric polynomials were obtained in [8]. Namely, w (r) n+m (y) admits a recurrence relation according to the family y j w (r+j) n (y) as follows: Setting y = 1 in (1.2), we have higher order geometric numbers w (r) n . The higher order geometric numbers and geometric numbers are connected with w (1) n = w n and the formula which was proved by a combinatorial method in [1,Theorem 2]. Here, n k is the Stirling number of the first kind ( [15]). Moreover, some congruence identities for the higher order geometric numbers can also be found in the recent work [11].
In this paper, dealing with two-variable geometric polynomials defined in [25] by we obtain new recurrence relations, an explicit formula, and a result generalizing (c.f. [8]) for higher order geometric polynomials. We particularly use the explicit formula to obtain an integral representation similar to (1.3) involving r-Bell polynomials, which are defined in [26] as where n+r k+r r are r-Stirling numbers of the second kind ( [3]). The resulting integral representation enables us to utilize some properties of r-Bell polynomials for higher order geometric polynomials. In particular, we evaluate the infinite sum in terms of higher order geometric polynomials, obtain an ordinary generating function for higher order geometric polynomials, introduce a new recurrence for w (r) n+m (y), and generalize (1.5). Moreover, we give an integral representation relating the higher order geometric polynomials and p-Bernoulli numbers, and express properties of p-Bernoulli numbers originated from those for the higher order geometric polynomials. We state and prove these results in Section 3, followed by Section 2, in which we summarize known results that we need in this paper. Section 4 is the last section of this paper where we prove congruences for higher order geometric polynomials and p-Bernoulli numbers using some of the results presented in Section 3. Particularly, we state a von Staudt-Clausen type congruence for p-Bernoulli numbers.

Preliminaries
The Stirling numbers of the first kind n k can be defined by means of x (x + 1) · · · (x + n − 1) = n k=0 n k x k or by the generating function (c.f. [9,15]). It follows from either of these definitions that We note the following special values which will be used in the sequel: Many properties of n k can be found in [9, pp. 214-219]. In particular, we have This equality can be used to obtain some congruences for n k . For example, if we take n = q, where q is a prime number, then The Stirling numbers of the second kind n k can be defined by means of or by the generating function (c.f. [9,15]). It follows from the generating function that We note the following well-known identity for n k for future reference: Performing the product of two generating functions for n k , we obtain the convolution formula ( [18]) Letting k = k 1 + k 2 and n = q, a prime number, we deduce that Stirling numbers have been generalized in many ways. One of them is called r-Stirling numbers (or weighted Stirling numbers). r-Stirling numbers of the second kind n k r can be defined by means of the generating function (see [3]) The Bernoulli numbers B n are defined by the generating function or by the equivalent recursion The first values are and B 2k+1 = 0 for k ≥ 1. The denominators of the Bernoulli numbers can be completely determined due to von Staudt-Clausen theorem: for any integer n ≥ 1, B 2n can be written as where A 2n is an integer and the sum runs over all the prime numbers such that (q − 1) |2n. It can be stated equivalently as We note that this classification is also valid for B 1 . Many generalizations of Bernoulli numbers appear in the literature. One generalization is the p-Bernoulli numbers B n,p , which are due to Rahmani [30], defined by means of the generating function p-Bernoulli numbers are related to Bernoulli numbers in that B n,0 = B n and and satisfy an explicit formula of the form

Recurrence Relations
From the generating function for higher order two-variable geometric polynomials (1.6) we have Then, it is obvious that Then, for x = 0, we conclude that a relationship between two-variable and single variable higher order geometric polynomials.
Proposition 3.1 For n ≥ 0 and r > 0, we have the following recurrence formulas: Proof.
Proof. We first note that Let x = x 1 + x 2 − 1 and r = r 1 + r 2 − 1. Then by (1.6), product of two infinite series, and formal differentiation under summation, we obtain that and d dt Equating coefficients of t n n! on both sides, we derive that n k=0 n k w Setting x 1 = r 1 , x 2 = r 2 and using (3.2) we obtain the convolution formula (3.5).
In the following theorem we give a new explicit expression for higher order geometric polynomials and numbers In particular Proof. Writing x = r in (1.6), employing the generalized binomial formula, and using the generating function of r-Stirling numbers (2.5) we have ∞ n=0 w (r) n (r; y) Comparing the coefficients of t n n! we obtain w (r) n (r; y) = n k=0 n + r k + r r (r) k y k .
Using (3.2) and replacing y with − (y + 1) , we reach the desired equation. Now, with use of Theorem 3.3, we connect higher order geometric polynomials and r-Bell polynomials in the following lemma which will be useful for the subsequent results.
Lemma 3.4 For every n ≥ 0 and every r > 0, we have the integral representation (3.7) Proof. By (1.7) we have Using (3.6) in the above yields the desired equation.
Higher order geometric polynomials are seen in the evaluation of the infinite series If we apply Lemma 3.4 to the Dobinski's formula for r-Bell polynomials we can evaluate a new infinite series in terms of higher order geometric polynomials.
Proof. We start by observing the ordinary generating function for r-Bell polynomials ([26, In light of the equation (3.7), this equation can be written as We then replace − (y + 1) with y and −t with t to obtain the desired equation. Now, we give an alternative representation for w (r) n+m (y), which also generalizes (1.5) in the following theorem. m + r k + r r y k ϕ n,r+k (y) in (3.7), we have which is equal to (3.8).
To prove (3.9), we use the formula y p ϕ n,r+p (y) = We note that it is also possible to derive this result by applying (1.3) and (3.7) in a formula given in [28,Eq. (9)]. Moreover, for r = 1, (3.9) can be written as which is also polynomial extension of (1.5). Replacing y by −y and integrating both sides with respect to y from 0 to 1, we have Then using (2.6), we obtain the following integral representation for p-Bernoulli numbers.
Theorem 3.9 For n ≥ 1 and p ≥ 0 The explicit formula (2.7) for p-Bernoulli numbers can be also deduced using this integral representation in (3.6).
Theorem 3.10 For n, p, m ≥ 0, we have For n, r ≥ 1 and p ≥ 0 we have Proof. Firstly, we replace y with −y in (3.9), multiply both sides by (1 − y) r−1 , and integrate with respect to y from 0 to 1. The result is n+k (−y) dy.

From (3.11) this equation turns into
Replacing n with n + 1 in the above equation completes the proof (3.13).
Applying the same method to the identity (3.8) gives (3.12).

Congruences
In this section we first consider congruences modulo a prime number q for higher order geometric polynomials. We start with two auxiliary results.
Lemma 4.1 Let q be an odd prime and y be an integer. Then, we have w q (y) ≡ y (mod q) .
Lemma 4.2 Let q be a prime and y be an integer. Then for all n ≥ 1, we have w q+n−1 (y) ≡ w n (y) (mod q) .
Proof. If q = 2, then by (1.1) since n 0 = 0 and n 1 = 1 for n > 0. Now, suppose that q is an odd prime and let n ≥ q − 1. Then again by (1.1) we write
If 1 ≤ n < q − 1, then we write since n k = 0 when k > n. Therefore, for n ≥ 1, w q+n−1 (y) ≡ w n (y) (mod q). We note that a more general result can be found in [2] for Fubini numbers. Theorem 4.3 Let q be an odd prime. If 1 + y is not a multiple of q, then w (q) q (y) ≡ 0 (mod q). Proof. We set p = q − 1 and n = q in (3.10) to obtain By Lemma 4.1 and Lemma 4.2, we find that since by (2.2) q k ≡ 0 (mod q) for 2 ≤ k ≤ q − 1 and w k (y) is an integer when y is an integer. The result now follows from Fermat's and Wilson's theorems.
It is obvious from (1.2) that if y is an integer which is a multiple of q, then w (r) n (y) ≡ 0 (mod q), since n k (r) k is an integer. We note that Theorem 4.3 is a special case which can be drawn from the following result.

Theorem 4.4
If y is an integer that is not a multiple of q, then w (r) n (y) ≡ 0 (mod q) for n ≥ 1 and r ≡ 0 (mod q).
Proof. Let r = tq for some integer t. By (1.2), we have we have the result.
Theorem 4.5 If y is an integer such that y and 1 + y are not multiples of an odd prime q, then w (r) q−1 (y) ≡ 0 (mod q) for r ≡ 1 (mod q). Proof. Let r = 1 + tq for some integer t. By (1.2), we have we deduce that It follows from (2.3) that q−1 (y) ≡ 1 + y − 1 + y q ≡ 0 (mod q) , and the result.
These results and their proofs are direct generalizations of the corresponding congruences for higher order geometric numbers given in [11,Corollary 4.2].
We conclude the study of congruences for higher order geometric polynomials by a similar result. Theorem 4.6 If y is an integer that is not a multiple of an odd prime q, then w (r) q+1 (y) ≡ 0 (mod q) for r ≡ 0 (mod q), and w (r) q+1 (y) ≡ −y (mod q) for r ≡ −1 (mod q).
Proof. For a prime q and nonnegative integer m, we have This result was given by Howard in [17], and can be easily verified by induction on m. It then follows that q+1 k ≡ 0 (mod q) for k = 3, 4, . . . , q, and q+1 from which the results follow.
In the rest of this section we consider congruences for p-Bernoulli numbers. In particular, the following theorem states a von Staudt-Clausen-type result for p-Bernoulli numbers.