# Recurrence formulas for the generalized Euler numbers [E.sup.(k)1.sub.2n]>.

Abstract In this paper, we prove some new recurrence formulas for
the generalized Euler numbers [E.sup.(k).sub.2n]).

Keywords The Euler numbers, the generalized Euler numbers, recurrence formula.

[section] 1. Introduction and results

For a real or complex parameter x, the generalized Enter numbers [E.sup.(x).sub.2n] are defined by the following generating functions (see [1]):

(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

or

(2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(If x is a nonnegative integer, then E[2.sup.n] are called Enter numbers of order x (see [2-4]).)

By (1), we have [E.sup.(x).sub.2n] = 0 (n [greater than or equal to] 0). The numbers -[2.sup.n] = E[2.sup.n] are the classical Euter numbers. By (1) or (2), we can get

(3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where k is a positive integer.

The Enter numbers E[2.sup.n] satisfy the recurrence relation

(4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

so we find [E.sub.2] = -1, [E.sub.4] = 5, [E.sub.6] = - 61, [E.sub.8] = 1385, [E.sub.10] = -50521, [E.sub.12] = 2702765, ....

By the mathematical induction, all the Euler numbers E0, Ez, E4,- .. are integers. By (3), we know that [E.sup.(k).sub.2n] is an integer.

In [5], Liu obtained some recurrence formulas for the generalized Enter numbers [E.sup.(k).sub.2n]

(5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and

(6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where n [greater than or equal to] 1, k [greater than or equal to] 0 are integers.

The main purpose of this paper is to prove some new recurrence formulas for the generalized Euler numbers. That is, we shall prove the following main conclusions.

Theorem 1. Let n [greater than or equal to] 1, k [greater than or equal to] 1, m [greater than or equal to] 1 are integers. Then we have

(7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where {[H.sup.(k).sub.2n]) (m)} I can be defined by the generating function

(8) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Taking k = 1, 2 in Theorem 1, we may immediately deduce the following

Corollary 1. Let n [greater than or equal to] 1, m [greater than or equal to] 1 are integers. Then

(9) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and

(10) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Curiously, we find that the following recurrences are special cases of Corollary 1.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Theorem 2. Let n [greater than or equal to] 1, k [greater than or equal to] 1 be integers. Then

(11) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(12) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Remark 1. By the inversion principle (see [6])

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

we may rephrase (11) as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

or

(13) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and

(14) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

[section] 2. Proof of the Theorems and Corollary

Proof of Theorem 1. Recall the generating function [(sec t).sup.k] in (2). Then, writing

[(sec t).sup.k] = [(sec(2m + 1)t).sup.k] [(cos(2m+1/).sup.k]/cost)

and forming the Abel convolution, we find that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Separating the term with j = n and solving, we obtain (7).

This completes the proof of Theorem 1.

Proof of Corollary 1. It suffices to find closed form of [H.sup.(k).sub.2n])(2m + 1) for k = 1, 2. We may rewrite the generating function (8) in the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

by straightforward formation. By multinational expansion we obtain

(15) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Hence we may immediately get

(16) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Then in the second identity, we divide the sums according to the parity of k and of k + 1 = [lambda].

Then we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Substituting (16) or (17) in (7) completes the proof of Corollary 1.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Proof of Theorem 2. Let [f.sub.k] (t) (2/[e.sup.t] = [e.sup.t]) Then in order to find [f.sub.k+1) (t), we are naturally led to differentiate it.

Since we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and it remains to form the Abel convolution of et and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(12) follows from the classifying the values of j modulo 2 and recalling that all odd indexed Euler numbers of order k are 0.

This completes the proof of Theorem 2.

(1) This work was supported by the Guangdong Provincial Natural Science Foundation (No. 05005928).

References

[1] G.D. Liu and W.P. Zhang, Applications of an explicit formula for the generalized Enter numbers, (Chinese) Acta Math. Sinica (English Series), 24(2008), No.2, 343-352.

[2] G.D. Liu, Summation and recurrence formula involving the central factorial numbers and zeta function, Appl. Math. Comput., 149(2004), No.1, 175-186.

[3] G.D. Liu, The generalized central factorial numbers and higher order Norlund Euler- Bernoulli polynomials, (Chinese) Acta Math. Sinica (Chin. Ser.), 44(2001), No.5, 933-946.

[4] G.D. Liu, Congruences for higher-order Euler numbers. Proc. Japan Acad. Ser.A, 82(2006), No.3, 30-33.

[5] G.D. Liu, Recurrence for generalized Enter numbers, Scientia Magna, 3(2007), No.l, 9-13.

[6] J. Riordan, Combinatorial Identities, Wiley, New York, 1968.

Guodong Liu

Department of Mathematics, Huizhou University,

Huizhou, Guangdong, 516015, P.R.China

Email: gdliu(Qpub.huizhou.gd.cn

Keywords The Euler numbers, the generalized Euler numbers, recurrence formula.

[section] 1. Introduction and results

For a real or complex parameter x, the generalized Enter numbers [E.sup.(x).sub.2n] are defined by the following generating functions (see [1]):

(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

or

(2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(If x is a nonnegative integer, then E[2.sup.n] are called Enter numbers of order x (see [2-4]).)

By (1), we have [E.sup.(x).sub.2n] = 0 (n [greater than or equal to] 0). The numbers -[2.sup.n] = E[2.sup.n] are the classical Euter numbers. By (1) or (2), we can get

(3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where k is a positive integer.

The Enter numbers E[2.sup.n] satisfy the recurrence relation

(4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

so we find [E.sub.2] = -1, [E.sub.4] = 5, [E.sub.6] = - 61, [E.sub.8] = 1385, [E.sub.10] = -50521, [E.sub.12] = 2702765, ....

By the mathematical induction, all the Euler numbers E0, Ez, E4,- .. are integers. By (3), we know that [E.sup.(k).sub.2n] is an integer.

In [5], Liu obtained some recurrence formulas for the generalized Enter numbers [E.sup.(k).sub.2n]

(5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and

(6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where n [greater than or equal to] 1, k [greater than or equal to] 0 are integers.

The main purpose of this paper is to prove some new recurrence formulas for the generalized Euler numbers. That is, we shall prove the following main conclusions.

Theorem 1. Let n [greater than or equal to] 1, k [greater than or equal to] 1, m [greater than or equal to] 1 are integers. Then we have

(7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where {[H.sup.(k).sub.2n]) (m)} I can be defined by the generating function

(8) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Taking k = 1, 2 in Theorem 1, we may immediately deduce the following

Corollary 1. Let n [greater than or equal to] 1, m [greater than or equal to] 1 are integers. Then

(9) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and

(10) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Curiously, we find that the following recurrences are special cases of Corollary 1.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Theorem 2. Let n [greater than or equal to] 1, k [greater than or equal to] 1 be integers. Then

(11) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(12) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Remark 1. By the inversion principle (see [6])

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

we may rephrase (11) as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

or

(13) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and

(14) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

[section] 2. Proof of the Theorems and Corollary

Proof of Theorem 1. Recall the generating function [(sec t).sup.k] in (2). Then, writing

[(sec t).sup.k] = [(sec(2m + 1)t).sup.k] [(cos(2m+1/).sup.k]/cost)

and forming the Abel convolution, we find that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Separating the term with j = n and solving, we obtain (7).

This completes the proof of Theorem 1.

Proof of Corollary 1. It suffices to find closed form of [H.sup.(k).sub.2n])(2m + 1) for k = 1, 2. We may rewrite the generating function (8) in the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

by straightforward formation. By multinational expansion we obtain

(15) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Hence we may immediately get

(16) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Then in the second identity, we divide the sums according to the parity of k and of k + 1 = [lambda].

Then we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Substituting (16) or (17) in (7) completes the proof of Corollary 1.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Proof of Theorem 2. Let [f.sub.k] (t) (2/[e.sup.t] = [e.sup.t]) Then in order to find [f.sub.k+1) (t), we are naturally led to differentiate it.

Since we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and it remains to form the Abel convolution of et and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(12) follows from the classifying the values of j modulo 2 and recalling that all odd indexed Euler numbers of order k are 0.

This completes the proof of Theorem 2.

(1) This work was supported by the Guangdong Provincial Natural Science Foundation (No. 05005928).

References

[1] G.D. Liu and W.P. Zhang, Applications of an explicit formula for the generalized Enter numbers, (Chinese) Acta Math. Sinica (English Series), 24(2008), No.2, 343-352.

[2] G.D. Liu, Summation and recurrence formula involving the central factorial numbers and zeta function, Appl. Math. Comput., 149(2004), No.1, 175-186.

[3] G.D. Liu, The generalized central factorial numbers and higher order Norlund Euler- Bernoulli polynomials, (Chinese) Acta Math. Sinica (Chin. Ser.), 44(2001), No.5, 933-946.

[4] G.D. Liu, Congruences for higher-order Euler numbers. Proc. Japan Acad. Ser.A, 82(2006), No.3, 30-33.

[5] G.D. Liu, Recurrence for generalized Enter numbers, Scientia Magna, 3(2007), No.l, 9-13.

[6] J. Riordan, Combinatorial Identities, Wiley, New York, 1968.

Guodong Liu

Department of Mathematics, Huizhou University,

Huizhou, Guangdong, 516015, P.R.China

Email: gdliu(Qpub.huizhou.gd.cn

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Author: | Liu, Guodong |
---|---|

Publication: | Scientia Magna |

Article Type: | Report |

Geographic Code: | 9CHIN |

Date: | Jan 1, 2008 |

Words: | 932 |

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