Hypergeometric Euler numbers

In this paper, we introduce the hypergeometric Euler number as an analogue of the hypergeometric Bernoulli number and the hypergeometric Cauchy number. We study several expressions and sums of products of hypergeometric Euler numbers. We also introduce complementary hypergeometric Euler numbers and give some characteristic properties. There are strong reasons why these hypergeometric numbers are important. The hypergeometric numbers have one of the advantages that yield the natural extensions of determinant expressions of the numbers, though many kinds of generalizations of the Euler numbers have been considered by many authors.


Introduction
Euler numbers E n are defined by the generating function One of the different definitions is (see e.g. [3]). Generalizations of one or other of these definitions have previously been studied. For example, one kind of poly-Euler numbers is a typical generalization, in the aspect of L-functions ( [21,22,23]). Other generalizations can be found in [5,16] and the reference therein. A different type of generalization is based upon hypergeometric functions. For N ≥ 1, define hypergeometric Bernoulli numbers B N,n (see [9,10,11] is the confluent hypergeometric function with (x) (n) = x(x + 1) · · · (x + n − 1) (n ≥ 1) and (x) (0) = 1. When N = 1, B n = B 1,n are classical Bernoulli numbers defined by t e t − 1 = ∞ n=0 B n t n n! .
In addition, define hypergeometric Cauchy numbers c N,n (see [12]) by Our generalization is different from the generalizations in [14] and the references therein. Now, for N ≥ 0 define hypergeometric Euler numbers E N,n (n = 0, 1, 2, . . . ) by 1 1 F 2 (1; N + 1, (2N + 1)/2; t 2 /4) where 1 F 2 (a; b, c; z) is the hypergeometric function defined by It is seen that When N = 0, then E n = E 0,n are classical Euler numbers defined in (1). In [18], the truncated Euler polynomial E m,n (x) is introduced as a generalization of the classical Euler polynomial E n (x). The concept is similar but without hypergeometric functions. We list the numbers E N,n for 0 ≤ N ≤ 6 and 0 ≤ n ≤ 12 in Table 1. From (3) we see that E N,n = 0 if n is odd. Similarly to poly-Euler numbers ( [21,22,23]), hypergeometric Euler numbers are rational numbers, though the classical Euler numbers are integers. We have an explicit expression of E N,n for each even n: Proof. The proof is done by induction for n. If n = 1, then .
Assume that the result is valid up to n − 1. Then by Proposition 1

Determinant expressions of hypergeometric numbers
These hypergeometric numbers have one of the advantages that yield the natural extensions of determinant expressions of the numbers, though many kinds of generalizations of the Euler numbers have been considered by many authors. By using Proposition 1 or the relation (4), we have a determinant expression of hypergeometric Euler numbers ( [13]). Proposition 2. The hypergeometric Euler numbers E N,2n (N ≥ 0, n ≥ 1) can be expressed as In 1875, Glaisher gave several interesting determinant expressions of numbers, including Bernoulli, Cauchy and Euler numbers. When N = 0, the determinant in Proposition (2) is reduced to a famous determinant expression of Euler numbers (cf. [6, p.52]): In [1], the hypergeometric Bernoulli numbers B N,n (N ≥ 1, n ≥ 1) can be expressed as When N = 1, we have a determinant expression of Bernoulli numbers ([6, p.53]): In [2], the hypergeometric Cauchy numbers c N,n (N ≥ 1, n ≥ 1) can be expressed as When N = 1, we have a determinant expression of Cauchy numbers ([6, p.50]): In [13], the complementary Euler numbers E n and their hypergeometric generalizations (defined below) have also determinant expressions.

Hasse-Teichmüller derivative
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. The Hasse-Teichmüller derivatives satisfy the product rule [24], the quotient rule [7] and the chain rule [8]. One of the product rules can be described as follows.
The quotient rules can be described as follows.
By using the Hasse-Teichmüller derivative of order n, we shall obtain some explicit expressions of the hypergeometric Euler numbers.
Another proof of Theorem 1. Put for simplicity. Note that Hence, by using Lemma 2 (7), we have We can express the hypergeometric Euler numbers also in terms of the binomial coefficients. In fact, by using Lemma 2 (8) instead of Lemma 2 (7) in the above proof, we obtain a little different expression from one in Theorem 1.
Proposition 3. For N ≥ 0 and even n ≥ 2, For example, when n = 4, we get

Some hypergeometric Euler numbers
If N = 1, we have the following relation between hypergeometric Euler numbers and Bernoulli numbers.
Theorem 2. For n ≥ 1 we have Proof. The result is clear for n = 0, 1 and odd numbers n. By using the following Lemma 3 and Proposition 1, we get the result.
On the other hand, Comparing the coefficients of x n , we get the result.

Sums of products of hypergeometric Euler numbers
It is known that First, let us consider the sums of products of hypergeometric Euler numbers: It is clear that In other words, they appear as numerators in the Maclaurin series of tan x: for simplicity again. Then by For further simplicity, we put for k = 1, 2, . . . , 2N with F (0) = F . Then, in general, we obtain for k = 1, 2, . . . , 2N Proposition 4. For k = 0, 1, . . . , 2N we have Proof. For k = 0, we get Assume that the result holds for some k ≥ 0. Then by (10) We introduce the complementary hypergeometric Euler numbers E N,n by as an analogue of (2). When n = 0, E n = E 0,n are the complementary Euler numbers defined by as an analogue of (1). In [18], they are called weighted Bernoulli numbers, but this naming means different in other literatures. Since by (9) we have 1 Since Comparing the coefficients, we obtain a result about the sums of products.
Using (11) and (12) again, we have Comparing the coefficients, we get a result about the sums of products for trinomial coefficients.

Complementary hypergeometric Euler numbers
By using the similar methods in previous sections, the complementary hypergeometric Euler numbers satisfy the recurrence relation for even n By using the Hasse-Teichmüller derivative or by proving by induction, we have Theorem 5. For N ≥ 0 and n ≥ 1 we have Some initial values of E N,n (n = 0, 2, 4, . . . ), we have , , , E N,n t n n! .
Hence, as an analogue of Theorem 3, we have the following.
We then have 1 we have the following result as an analogue of Theorem 4. Theorem 7. For N ≥ 1 and n ≥ 0, One can continue to obtain the sum of four or more products, though the results seem to become more complicated.

Applications by the Trudi's formula
We shall use the Trudi's formula to obtain different explicit expressions for the hypergeometric Euler numbers E N,n .
In addition, there exists the following inversion formula (see, e.g. [17]), which is based upon the relation:
By applying these lemmata to Proposition 2, we obtain an explicit expression for the hypergeometric Euler numbers E N,n .
Similarly, by the results in [13], after applying Lemmata 4 and 5, we have a new expression of the complementary hypergeometric Euler numbers E N,n .