Discrete Dynamics in Nature and Society

Discrete Dynamics in Nature and Society / 2009 / Article

Research Article | Open Access

Volume 2009 |Article ID 605313 | 6 pages | https://doi.org/10.1155/2009/605313

A Recurrence Formula for 𝐷 Numbers 𝐷(2π‘›βˆ’1)2𝑛

Academic Editor: Binggen Zhang
Received28 Jun 2009
Accepted16 Oct 2009
Published19 Nov 2009

Abstract

we establish a recurrence formula for 𝐷 numbers 𝐷(2π‘›βˆ’1)2𝑛. A generating function for 𝐷 numbers 𝐷(2π‘›βˆ’1)2𝑛 is also presented.

1. Introduction and Results

The Bernoulli polynomials 𝐡𝑛(π‘˜)(π‘₯) of order π‘˜, for any integer π‘˜, may be defined by (see [1–4])

ξ‚€π‘‘π‘’π‘‘ξ‚βˆ’1π‘˜π‘’π‘₯𝑑=βˆžξ“π‘›=0𝐡𝑛(π‘˜)(𝑑π‘₯)𝑛𝑛!,|𝑑|<2πœ‹.(1.1)

The numbers 𝐡𝑛(π‘˜)=𝐡𝑛(π‘˜)(0) are the Bernoulli numbers of order π‘˜, 𝐡𝑛(1)=𝐡𝑛 are the ordinary Bernoulli numbers (see [2, 5]). By (1.1), we can get (see [4, page 145])

𝑑𝐡𝑑π‘₯𝑛(π‘˜)(π‘₯)=𝑛𝐡(π‘˜)π‘›βˆ’1(π‘₯),(1.2)𝐡𝑛(π‘˜+1)(π‘₯)=π‘˜βˆ’π‘›π‘˜π΅π‘›(π‘˜)𝑛(π‘₯)+(π‘₯βˆ’π‘˜)π‘˜π΅(π‘˜)π‘›βˆ’1(π‘₯),(1.3)𝐡𝑛(π‘˜+1)(π‘₯+1)=𝑛π‘₯π‘˜π΅(π‘˜)π‘›βˆ’1(π‘₯)βˆ’π‘›βˆ’π‘˜π‘˜π΅π‘›(π‘˜)(π‘₯),(1.4) where π‘›βˆˆβ„•, with β„• being the set of positive integers.

The numbers 𝐡𝑛(𝑛) are called the NΓΆrlund numbers (see [2, 4, 6]). A generating function for the NΓΆrlund numbers 𝐡𝑛(𝑛) is (see [4, page 150])

𝑑=(1+𝑑)log(1+𝑑)βˆžξ“π‘›=0𝐡𝑛(𝑛)𝑑𝑛.𝑛!(1.5)

The 𝐷 numbers 𝐷(π‘˜)2𝑛 may be defined by (see [4, 7, 8])

(𝑑csc𝑑)π‘˜=βˆžξ“π‘›=0(βˆ’1)𝑛𝐷(π‘˜)2𝑛𝑑2𝑛(2𝑛)!,|𝑑|<πœ‹.(1.6)

By (1.1), (1.6), and note that csc𝑑=2𝑖/(π‘’π‘–π‘‘βˆ’π‘’βˆ’π‘–π‘‘) (where 𝑖2=βˆ’1), we can get

𝐷(π‘˜)2𝑛=4𝑛𝐡(π‘˜)2π‘›ξ‚€π‘˜2.(1.7)

Taking π‘˜=1,2 in (1.7), and note that 𝐡(1)2𝑛(1/2)=(21βˆ’2π‘›βˆ’1)𝐡2𝑛, 𝐡(2)2𝑛(1)=(1βˆ’2𝑛)𝐡2𝑛 (see [4, page 22, page 145]), we have

𝐷(1)2𝑛=ξ€·2βˆ’22𝑛𝐡2𝑛,𝐷(2)2𝑛=4𝑛(1βˆ’2𝑛)𝐡2𝑛.(1.8)

The 𝐷 numbers 𝐷(π‘˜)2𝑛 satisfy the recurrence relation (see [7])

𝐷(π‘˜)2𝑛=(2π‘›βˆ’π‘˜+2)(2π‘›βˆ’π‘˜+1)(π·π‘˜βˆ’2)(π‘˜βˆ’1)(π‘˜βˆ’2)2π‘›βˆ’2𝑛(2π‘›βˆ’1)(π‘˜βˆ’2)π·π‘˜βˆ’1(π‘˜βˆ’2)2π‘›βˆ’2.(1.9)

By (1.9), we may immediately deduce the following (see [4, page 147]):

𝐷(2𝑛+1)2𝑛=(βˆ’1)𝑛(2𝑛)!4𝑛𝑛ξƒͺ2𝑛,𝐷(2𝑛+2)2𝑛=(βˆ’1)𝑛4𝑛2𝑛+1(𝑛!)2,(1.10)𝐷(2𝑛+3)2𝑛=(βˆ’1)𝑛(2𝑛)!2β‹…42𝑛ξƒͺξ‚΅12𝑛+2𝑛+11+32+1521+β‹―+(2𝑛+1)2ξ‚Ά.(1.11)

The numbers 𝐷(2𝑛)2𝑛 are called the 𝐷-NΓΆrlund numbers that satisfy the recurrence relation (see [7])

𝑛𝑗=0(βˆ’1)𝑗4𝑗𝑗ξƒͺ𝐷(2𝑗+1)2𝑗(2π‘›βˆ’2𝑗)2π‘›βˆ’2𝑗=(2π‘›βˆ’2𝑗)!(βˆ’1)𝑛4𝑛𝑛ξƒͺ,2𝑛(1.12) so we find 𝐷0(0)=1,𝐷2(2)=βˆ’2/3,𝐷4(4)=88/15,𝐷6(6)=βˆ’3056/21,𝐷8(8)=319616/45,𝐷(10)10=βˆ’18940160/33,….

A generating function for the 𝐷-NΓΆrlund numbers 𝐷(2𝑛)2𝑛 is (see [7])

π‘‘βˆš1+𝑑2ξ‚€βˆšlog𝑑+1+𝑑2=βˆžξ“π‘›=0𝐷(2𝑛)2𝑛𝑑2𝑛(2𝑛)!,|𝑑|<1.(1.13)

These numbers 𝐷(2𝑛)2𝑛 and 𝐷(2π‘›βˆ’1)2𝑛 have many important applications. For example (see [4, page 246])

ξ€œ0πœ‹/2sin𝑑𝑑𝑑𝑑=βˆžξ“π‘›=0(βˆ’1)𝑛𝐷(2𝑛)2𝑛,ξ€œ(2𝑛+1)!0πœ‹/2sinπ‘‘π‘‘πœ‹π‘‘π‘‘=2βˆžξ“π‘›=0(βˆ’1)𝑛+1𝐷(2π‘›βˆ’1)2𝑛22𝑛(2π‘›βˆ’1)(𝑛!)2,2πœ‹=βˆžβˆ‘π‘›=0(βˆ’1)𝑛+1𝐷(2π‘›βˆ’1)2𝑛(.2π‘›βˆ’1)(2𝑛)!(1.14)

The main purpose of this paper is to prove a recurrence formula for 𝐷 numbers 𝐷(2π‘›βˆ’1)2𝑛 and to obtain a generating function for 𝐷 numbers 𝐷(2π‘›βˆ’1)2𝑛. That is, we will prove the following main conclusion.

Theorem 1.1. Let π‘›βˆˆβ„•. Then 𝑛𝑗=1ξƒͺ2𝑛2𝑗(βˆ’1)π‘—βˆ’14π‘—βˆ’1((π‘—βˆ’1)!)2𝐷(2π‘›βˆ’1βˆ’2𝑗)2π‘›βˆ’2𝑗=(βˆ’1)π‘›βˆ’12(2𝑛)!4𝑛ξƒͺ,2π‘›βˆ’2π‘›βˆ’1(1.15) so one finds 𝐷2(1)=βˆ’1/3,𝐷4(3)=17/5,𝐷6(5)=βˆ’1835/21,𝐷8(7)=195013/45,𝐷(9)10=βˆ’3887409/11,….

Theorem 1.2. Let 𝑑 be a complex number with |𝑑|<1. Then βˆžξ“π‘›=0𝐷(2π‘›βˆ’1)2𝑛𝑑2𝑛=1(2𝑛)!√1+𝑑2ξƒ©π‘‘βˆšlog(𝑑+1+𝑑2)ξƒͺ2.(1.16)

2. Proof of the Theorems

Proof of Theorem 1.1. Note the identity (see [4, page 203]) 𝐡(π‘˜)2π‘›ξ‚€π‘˜π‘₯+2=𝑛𝑗=0ξƒͺ𝐷2𝑛2𝑗(π‘˜βˆ’2𝑗)2π‘›βˆ’2𝑗22π‘›βˆ’2𝑗π‘₯2ξ€·π‘₯2βˆ’12π‘₯ξ€Έξ€·2βˆ’22ξ€Έβ‹―ξ€·π‘₯2βˆ’(π‘—βˆ’1)2ξ€Έ,(2.1) we have 𝐡(π‘˜)2𝑛(π‘₯+π‘˜/2)βˆ’π΅(π‘˜)2𝑛(π‘˜/2)π‘₯2=𝑛𝑗=1ξƒͺ𝐷2𝑛2𝑗(π‘˜βˆ’2𝑗)2π‘›βˆ’2𝑗22π‘›βˆ’2𝑗π‘₯2βˆ’12π‘₯ξ€Έξ€·2βˆ’22ξ€Έβ‹―ξ€·π‘₯2βˆ’(π‘—βˆ’1)2ξ€Έ.(2.2) Therefore, limπ‘₯β†’0𝐡(π‘˜)2𝑛(π‘₯+π‘˜/2)βˆ’π΅(π‘˜)2𝑛(π‘˜/2)π‘₯2=𝑛𝑗=1ξƒͺ𝐷2𝑛2𝑗(π‘˜βˆ’2𝑗)2π‘›βˆ’2𝑗22π‘›βˆ’2𝑗(βˆ’1)π‘—βˆ’1((π‘—βˆ’1)!)2.(2.3) By (2.3) and (1.2), we have limπ‘₯β†’02𝑛(2π‘›βˆ’1)𝐡(π‘˜)2π‘›βˆ’2(π‘₯+π‘˜/2)2=𝑛𝑗=1ξƒͺ𝐷2𝑛2𝑗(π‘˜βˆ’2𝑗)2π‘›βˆ’2𝑗22π‘›βˆ’2𝑗(βˆ’1)π‘—βˆ’1((π‘—βˆ’1)!)2.(2.4) That is, 𝑛(2π‘›βˆ’1)𝐡(π‘˜)2π‘›βˆ’2ξ‚€π‘˜2=𝑛𝑗=1ξƒͺ𝐷2𝑛2𝑗(π‘˜βˆ’2𝑗)2π‘›βˆ’2𝑗22π‘›βˆ’2𝑗(βˆ’1)π‘—βˆ’1((π‘—βˆ’1)!)2.(2.5) By (2.5) and (1.7), we have 𝐷(π‘˜)2π‘›βˆ’2=1𝑛(2π‘›βˆ’1)𝑛𝑗=1ξƒͺ2𝑛2𝑗(βˆ’1)π‘—βˆ’14π‘—βˆ’1((π‘—βˆ’1)!)2𝐷(π‘˜βˆ’2𝑗)2π‘›βˆ’2𝑗.(2.6) Setting π‘˜=2π‘›βˆ’1 in (2.6), and note (1.10), we immediately obtain Theorem 1.1. This completes the proof of Theorem 1.1.

Remark 2.1. Setting π‘˜=2𝑛 in (2.6), and note (1.10), we may immediately deduce the following recurrence formula for 𝐷-NΓΆrlund numbers 𝐷(2𝑛)2𝑛: 𝑛𝑗=1ξƒͺ2𝑛2𝑗(βˆ’1)𝑗4𝑗((π‘—βˆ’1)!)2𝐷(2π‘›βˆ’2𝑗)2π‘›βˆ’2𝑗=(βˆ’1)𝑛𝑛4𝑛((π‘›βˆ’1)!)2(π‘›βˆˆβ„•).(2.7)

Proof of Theorem 1.2. Note the identity (see [9]) βˆžξ“π‘›=0(βˆ’1)𝑛4𝑛(𝑛!)2𝑑2𝑛=1(2𝑛)!1+𝑑2𝑑1βˆ’βˆš1+𝑑2ξ‚€βˆšlog𝑑+1+𝑑2ξƒͺ,(2.8) where |𝑑|<1. We have βˆžξ“π‘›=0(βˆ’1)𝑛4𝑛(𝑛!)2𝑑2𝑛+2=1(2𝑛+2)!2ξ‚€ξ‚€βˆšlog𝑑+1+𝑑22,(2.9) That is, βˆžξ“π‘›=1(βˆ’1)π‘›βˆ’14π‘›βˆ’1((π‘›βˆ’1)!)2𝑑2𝑛=1(2𝑛)!2ξ‚€ξ‚€βˆšlog𝑑+1+𝑑22.(2.10)
On the other hand, βˆžξ“π‘›=1(βˆ’1)π‘›βˆ’12(2𝑛)!4𝑛ξƒͺ𝑑2π‘›βˆ’2π‘›βˆ’12𝑛=1(2𝑛)!2βˆžξ“π‘›=0(βˆ’1)𝑛4𝑛𝑛ξƒͺ𝑑2𝑛2𝑛+2=𝑑22√1+𝑑2.(2.11) Thus, by (2.10), (2.11), and Theorem 1.1, we have βˆžξ“π‘›=1(βˆ’1)π‘›βˆ’14π‘›βˆ’1((π‘›βˆ’1)!)2𝑑2𝑛(2𝑛)!βˆžξ“π‘›=1𝐷(2π‘›βˆ’1)2𝑛𝑑2𝑛=(2𝑛)!βˆžξ“π‘›=1(βˆ’1)π‘›βˆ’12(2𝑛)!4𝑛ξƒͺ𝑑2π‘›βˆ’2π‘›βˆ’12𝑛.(2𝑛)!(2.12) That is, 12ξ‚€ξ‚€βˆšlog𝑑+1+𝑑22βˆžξ“π‘›=1𝐷(2π‘›βˆ’1)2𝑛𝑑2𝑛=𝑑(2𝑛)!22√1+𝑑2.(2.13) By (2.13), and note that lim𝑑→0π‘‘ξ‚€βˆšlog𝑑+1+𝑑2=1,𝐷0(βˆ’1)=1,(2.14) we immediately obtain Theorem 1.2. This completes the proof of Theorem 1.2.

Acknowledgment

This work was Supported by the Guangdong Provincial Natural Science Foundation (no. 8151601501000002).

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Copyright © 2009 Guodong Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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