# Recurrences for generalized Euler numbers (1).

Abstract In this paper, we establish some recurrence formulas for
generalized Euler numbers.

Keywords Euler numbers, generalized Euler numbers, Bernoulli numbers, recurrence formula.

[section] 1. Introduction and results

For an integer k, the generalized Euler numbers [E.sup.(k).sub.2n] and the generalized Bernoulli numbers [B.sup.(k).sub.n] are defined by the following generating functions (see, for details, [1], [2], [3] and [4]):

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

and

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

respectively. Clearly, we have

[E.sup.(1).sub.2n] = [E.sub.2n] and [B.sup.(1).sub.n] = [B.sub.n] (n [member of] [[??].sub.0] := [??] [union] {0}) (3)

in terms of the classical Euler numbers [E.sub.2n] and the classical Bernoulli numbers [B.sub.n], [??] being the set of positive integers. The Euler numbers [E.sub.2n] and the Bernoulli numbers [B.sub.n] satisfy

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

and

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

By (1) and (2), we have

[B.sub.2n] = 2n / [2.sup.2n]([2.sup.2n] - 1)[E.sup.(2).sub.2n-2] (n [member of] [??]). (6)

Numerous interesting (and useful) properties and relationships involving each of these families of numbers can be found in many books and tables (see [5], [6] and [7]). The main purpose of this paper is to establish some recurrence formulas for generalized Euler numbers. That is, we shall prove the following main conclusion.

Theorem 1. Let n [member of] [??]; k [member of] [[??].sub.0]. Then

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

Remark 1. Setting k = 0 in (7), we immediately obtain (4).

Theorem 2. Let n [member of] [??]; k [member of] [[??].sub.0]. Then

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

Remark 2. Setting k = 0 in (8), we can get

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

By (9) and (6), we have

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

[section] 2. Some lemmas

Lemma 1. Let k [member of] [[??].sub.0]. Then

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

Proof.

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

i.e.

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

Lemma 2. Let k [greater than or equal to] 0 be integers, then

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

Proof.

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

i.e.

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

Remark 3. Taking x = 0 in Lemma 1 and Lemma 2, we can get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (17)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (18)

[section] 3. Proof of the theorems

Proof of Theorem 1. By Lemma 1 and (1), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (19)

and comparing the coefficient of x2n on both sides of (19), we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (20)

i.e.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (21)

By (3.3) and (2.7), we immediately obtain Theorem 1. This completes the proof of Theorem 1. Proof of Theorem 2. By Lemma 2 and (1), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (22)

and comparing the coefficient of [x.sup.2n] on both sides of (22), we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (23)

i.e.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (24)

By (24) and (18), we immediately obtain Theorem 2. This completes the proof of Theorem 2.

(1) This work is supported by Guangdong Provincial Natural Science Foundation of China (05005928).

Received Jan. 23, 2007

References

[1] Liu, G. D., Srivastava, H. M., Explicit formulas for the Norlund polynomials [B.sup.(x).sub.n] and [b.sup.(x).sub.n], Comput. Math. Appl. 51(2006), 1377-1384.

[2] Liu, G. D., Zhang, W. P., Applications of an explicit formula for the generalized Euler numbers, Acta Math. Sinica (English Series) (Chinese), to appear.

[3] Liu, G. D., Congruences for higher-order Euler numbers, Proc. Japan Acad. (Ser.A), 82(2006), 30-33.

[4] Liu, G. D., Summation and recurrence formula involving the central factorial numbers and zeta function, Appl. Math. Comput. 149(2004), 175-186.

[5] Erdelyi, A., Magnus, W., Oberhettinger, F., Tricomi, F. G., Higher Transcendental Functions, Vol. I, McGraw-Hill Book Company, 1953.

[6] Luke, Y. L., The Special Functions and Their Approximations, Vol. I, Academic Press, New York and London, 1969.

[7] Srivastava, H. M., Choi, J., Series Associated with the Zeta and Related Functions, Kluwer Academic Publishers, Dordrecht, Boston and London, 2001.

Guodong Liu ([dagger]) and Hui Li ([double dagger])

([dagger]). Department of Mathematics, Huizhou University, Huizhou, Guangdong, P. R. China e-mail: gdliu@pub.huizhou.gd.cn.

([double dagger]). Department of Mathematics, Jiaying University, Meizhou, Guangdong, P.R. China

Keywords Euler numbers, generalized Euler numbers, Bernoulli numbers, recurrence formula.

[section] 1. Introduction and results

For an integer k, the generalized Euler numbers [E.sup.(k).sub.2n] and the generalized Bernoulli numbers [B.sup.(k).sub.n] are defined by the following generating functions (see, for details, [1], [2], [3] and [4]):

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

and

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

respectively. Clearly, we have

[E.sup.(1).sub.2n] = [E.sub.2n] and [B.sup.(1).sub.n] = [B.sub.n] (n [member of] [[??].sub.0] := [??] [union] {0}) (3)

in terms of the classical Euler numbers [E.sub.2n] and the classical Bernoulli numbers [B.sub.n], [??] being the set of positive integers. The Euler numbers [E.sub.2n] and the Bernoulli numbers [B.sub.n] satisfy

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

and

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

By (1) and (2), we have

[B.sub.2n] = 2n / [2.sup.2n]([2.sup.2n] - 1)[E.sup.(2).sub.2n-2] (n [member of] [??]). (6)

Numerous interesting (and useful) properties and relationships involving each of these families of numbers can be found in many books and tables (see [5], [6] and [7]). The main purpose of this paper is to establish some recurrence formulas for generalized Euler numbers. That is, we shall prove the following main conclusion.

Theorem 1. Let n [member of] [??]; k [member of] [[??].sub.0]. Then

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

Remark 1. Setting k = 0 in (7), we immediately obtain (4).

Theorem 2. Let n [member of] [??]; k [member of] [[??].sub.0]. Then

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

Remark 2. Setting k = 0 in (8), we can get

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

By (9) and (6), we have

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

[section] 2. Some lemmas

Lemma 1. Let k [member of] [[??].sub.0]. Then

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

Proof.

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

i.e.

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

Lemma 2. Let k [greater than or equal to] 0 be integers, then

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

Proof.

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

i.e.

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

Remark 3. Taking x = 0 in Lemma 1 and Lemma 2, we can get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (17)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (18)

[section] 3. Proof of the theorems

Proof of Theorem 1. By Lemma 1 and (1), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (19)

and comparing the coefficient of x2n on both sides of (19), we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (20)

i.e.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (21)

By (3.3) and (2.7), we immediately obtain Theorem 1. This completes the proof of Theorem 1. Proof of Theorem 2. By Lemma 2 and (1), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (22)

and comparing the coefficient of [x.sup.2n] on both sides of (22), we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (23)

i.e.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (24)

By (24) and (18), we immediately obtain Theorem 2. This completes the proof of Theorem 2.

(1) This work is supported by Guangdong Provincial Natural Science Foundation of China (05005928).

Received Jan. 23, 2007

References

[1] Liu, G. D., Srivastava, H. M., Explicit formulas for the Norlund polynomials [B.sup.(x).sub.n] and [b.sup.(x).sub.n], Comput. Math. Appl. 51(2006), 1377-1384.

[2] Liu, G. D., Zhang, W. P., Applications of an explicit formula for the generalized Euler numbers, Acta Math. Sinica (English Series) (Chinese), to appear.

[3] Liu, G. D., Congruences for higher-order Euler numbers, Proc. Japan Acad. (Ser.A), 82(2006), 30-33.

[4] Liu, G. D., Summation and recurrence formula involving the central factorial numbers and zeta function, Appl. Math. Comput. 149(2004), 175-186.

[5] Erdelyi, A., Magnus, W., Oberhettinger, F., Tricomi, F. G., Higher Transcendental Functions, Vol. I, McGraw-Hill Book Company, 1953.

[6] Luke, Y. L., The Special Functions and Their Approximations, Vol. I, Academic Press, New York and London, 1969.

[7] Srivastava, H. M., Choi, J., Series Associated with the Zeta and Related Functions, Kluwer Academic Publishers, Dordrecht, Boston and London, 2001.

Guodong Liu ([dagger]) and Hui Li ([double dagger])

([dagger]). Department of Mathematics, Huizhou University, Huizhou, Guangdong, P. R. China e-mail: gdliu@pub.huizhou.gd.cn.

([double dagger]). Department of Mathematics, Jiaying University, Meizhou, Guangdong, P.R. China

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Author: | Liu, Guodong; Li, Hui |
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Publication: | Scientia Magna |

Geographic Code: | 9CHIN |

Date: | Jan 1, 2007 |

Words: | 769 |

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