Several properties of hypergeometric Bernoulli numbers

In this paper, we give the determinant expressions of the hypergeometric Bernoulli numbers, and some relations between the hypergeometric and the classical Bernoulli numbers which include Kummer's congruences. By applying Trudi's formula, we have some different expressions and inversion relations. We also determine explicit forms of convergents of the generating function of the hypergeometric Bernoulli numbers, from which several identities for hypergeometric Bernoulli numbers are given.

In addition, define hypergeometric Bernoulli polynomials B N,n (z) ( [10]) by the generating function Many kinds of generalizations of the Bernoulli numbers have been considered by many authors. For example, Poly-Bernoulli number, multiple Bernoulli numbers, Apostol Bernoulli numbers, multi-poly-Bernoulli numbers, degenerated 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.
In [14], some determinant expressions of hypergeometric Cauchy numbers are considered. In this paper, we shall give the similar determinant expression of hypergeometric Bernoulli numbers and their generalizations. Then we study some relations between the hypergeometric Bernoulli numbers and the classical Bernoulli numbers which include Kummer's congruences. Furthermore, by applying Trudi's formula, we also have some different expressions and inversion relations. We also determine explicit forms of convergents of the generating function of the hypergeometric Bernoulli numbers, from which several identities for hypergeometric Bernoulli numbers are given.

Some basic properties of hypergeometric Bernoulli numbers
From the definition (1), we have Hence, for n ≥ 1, we have the following. Remark. When N = 1, we have a famous identity for Bernoulli numbers. If Bernoulli numbers B n are defined by x n n! , then it holds that n m=0 n + 1 m B m = n + 1 (n ≥ 1) .

By using Proposition 1 or
with B N,0 = 1 (N ≥ 1), some values of B N,n (0 ≤ n ≤ 9) are explicitly given by the following. , , In general, we have an explicit expression of B N,n .
Proposition 2. For N, n ≥ 1, we have Remark. In the later section about Trudi's formula, we see a different expression of B N,n in Corollary 2. Further, an inversion expression can be obtained: where n t1,...,t k = n! t1!···t k ! are the multinomial coefficients. Proof of Proposition 2. The proof can be done by induction on n. Here, we shall prove directly by using the generating function. From the definition (1), we have The proposition immediately follows by comparing coefficients of both sides.
We also have a different expression of B N,n with binomial coefficients. The proof is similar to that of Proposition 2 and omitted.

Analog of Kummer's congruence
Let p be a prime number, and ν ≥ 0 be an integer. If m and n are positive even integers with m ≡ n (mod (p − 1)p ν ) and m, n ≡ 0 (mod p − 1), then we have and this is called Kummer's congruence ([21, Corollary 5.14]). We get the similar congruence for the hypergeometric Bernoulli numbers B N,n if N is p-adically close enough to 1, that is, ord p (N − 1) is enough large. (1 + k)! B n (mod p t ), where t = ord p (N − 1).
Proof. In the case n = 0, the assertion is trivial. Assume that the result is true up to n − 1. By Proposition 1, we have From this lemma, we have the following corollary.

Determinant expressions
Remark. When N = 1, we have a determinant expression of Bernoulli numbers ([3, p.53]): Proof of Theorem 1. This Theorem is a special case of Theorem 2.

A relation between B N,n and B N −1,n
In this section, we show the following relation between B N,n and B N −1,n .
Lemma 2. For N ≥ 2 and n ≥ 1, we have Proof. From the derivative of (1), we have Therefore, we have and hence, Proof of Proposition 5. We give the proof by induction for n. In the case n = 1, the assertion means B N,1 = N N +1 B N −1,1 , and this equality follows from Assume that the assertion holds up to n − 1. By Lemma 2, we have

Multiple hypergeometric Bernoulli numbers
For positive integers N and r, define the higher order hypergeometric Bernoulli numbers B (r) N,n ( [11,17]) by the generating function The higher order hypergeometric Bernoulli polynomials B (r) Hence, as a generalization of Proposition (1), for n ≥ 1, we have the following.
By using Proposition 6 or with B We shall introduce the Hasse-Teichmüller derivative in order to prove Proposition 7 easily. Let F be a field of any characteristic, F[[z]] the ring of formal power series in one variable z, and F((z)) the field of Laurent series in z. Let n be a nonnegative integer. We define the Hasse-Teichmüller derivative H (n) of order n by where R is an integer and c m ∈ F for any m ≥ R. Note that m n = 0 if m < n. The Hasse-Teichmüller derivatives satisfy the product rule [18], the quotient rule [4] and the chain rule [5]. One of the product rules can be described as follows.
The quotient rules can be described as follows.
Lemma 4. For f ∈ F[[z]]\{0} and n ≥ 1, we have = n k=1 n + 1 k + 1 Since by the product rule of the Hasse-Teichmüller derivative in Lemma 3, we get Hence, by the quotient rule of the Hasse-Teichmüller derivative in Lemma 4 (9), we have .
where M r (e) are given in (8).
Remark. When r = 1 in Theorem 2, we have the result in Theorem 1.
Proof. For simplicity, put A N,n /n!. Then, we shall prove that for any n ≥ 1 When n = 1, (11) is valid because Assume that (11) is valid up to n − 1. Notice that by (7), we have Thus, by expanding the first row of the right-hand side (11), it is equal to Note that A where t1+···+tm t1,...,tm are the multinomial coefficients.
In addition, there exists the following inversion formula (see, e.g. [13]), which is based upon the relation: Lemma 7. If {α n } n≥0 is a sequence defined by α 0 = 1 and

Moreover, if
From Trudi's formula, it is possible to give the combinatorial expression By applying these lemmata to Theorem 2, we obtain an explicit expression for the generalized hypergeometric Bernoulli numbers B (r) where M r (e) are given in (8). Moreover, When r = 1 in Theorem 3, we have an explicit expression for the numbers B N,n .

Continued fractions of hypergeometric Bernoulli numbers
In [1,12] by studying the convergents of the continued fraction of some identities of Bernoulli numbers are obtained. In this section, the n-th convergent of the generating function of hypergeometric Bernoulli numbers is explicitly given. As an application, we give some identities of hypergeometric Bernoulli numbers in terms of binomial coefficients. The generating function on the left-hand side of (1) can be expanded as a continued fraction (Cf. [20, (91.2)]). Its n-th convergent P n (x)/Q n (x) (n ≥ 0) is given by the recurrence relation P n (x) = a n (x)P n−1 (x) + b n (x)P n−2 (x) (n ≥ 2), with initial values where for n ≥ 1, a n (x) = N + n, b 2n (x) = nx and b 2n+1 (x) = −(N + n)x.
We have explicit expressions of both the numerator and the denominator of the n-th convergent of (12).
Theorem 4. For n ≥ 1, we have Remark. Here we use the convenient values n k = 0 (0 ≤ n < k), −1 0 = 1 and recognize the empty product as 1. Otherwise, we should write Q 2n (x) as If we use the unsinged Stirling numbers of the first kind n k , which generating function is given by we can express the products as Proof of Theorem 4. The proof is done by induction on n. It is easy to see that for n = 0 we have P 0 (x) = Q 0 (x) = 1, and for n = 1 we have P 1 (x) = (N +1)−x and Q 1 (x) = N + 1. Assume that the results hold up to n − 1(≥ 2). Then by using the recurrence relation in (14) ( we get Next, (N + l) we get Concerning Q n (x), Since N + 2n = (N + 2n − j) + j, Similarly, (N + l) (N + l) we have (N + 2n + 1)Q 2n (x) − (N + n)xQ 2n−1 (x)

Some more identities of hypergeometric Bernoulli numbers
Since P 2n−1 (x), P 2n (x) and Q 2n (x) are the polynomials with degree n and Q 2n−1 (x) is the polynomial with degree n − 1, by the approximation property of the continued fraction, we have the following.
Lemma 8. Let P n (x)/Q n (x) denote the n-th convergent of the continued fraction expansion of (12). Then we have for n ≥ 0 By this approximation property, the coefficients x j (0 ≤ j ≤ n) of are nullified. By Theorem 4, Similarly, since (N + l) (0 ≤ h ≤ n); 0 (h > n) . (h > n) (15) and min{h,n} In particular, when N = 1, we have the relations for the classical Bernoulli numbers.
we can write (18) as Here the empty summation is recognized as 0, as usual.