Aq-Analogof the Weideman's Formula
2011-12-23ZHENGDeyinCHENGuang
ZHENG De-yin,CHEN Guang
(College of Science,Hangzhou Normal University,Hangzhou 310036,China)
Aq-Analogof the Weideman's Formula
ZHENG De-yin,CHEN Guang
(College of Science,Hangzhou Normal University,Hangzhou 310036,China)
1 Introduction
Recently,Weideman's formula[1,Eq.(20)]called one of the hardest challenge identities:
is closely concerned,where the harmonic numbers Hnand the second order harmonic number H(2)nare defined by
respectively.Schneider[2,Eq.(16)](cf.[3,Eq.(12)]also)proved the formula(1)via computer algebra package Sigma,while Chu proved it using partial fraction method in[4,Eq.(6)]and hypergeometric series method in[5,Eq.(3)].The main purpose of this paper is to find q-analogs of Weideman's formula by means of partial fraction decomposition.
We use the standard notation on q-series.The q-shifted factorial(a;q)nis defined by
The q-binomial coefficient,or the Gauss coefficient,is given by
The paper investigated the decomposition of a class of rational function by partial fraction method,established a generalized identity about q-harmonic numbers,and obtained twelve striking q-like-Weideman formulas from twelve special cases of this general identity.
q-binomial coefficients;q-harmonic numbers;algebraic identities
With the above preparations,we can establish the following general q-algebraicidentity.
2 Partial fraction decompositions
Theorem 1 Let xbe an indeterminate and napositive integer.For any polynomial Q(x)of degree≤2+3n,we have
Multiplying by xacross equation(4)and then letting x→+∞,we can obtain immediately the following general identity on q-binomial-harmonic number.
Theorem 2 Let n be a positive integer.For any polynomial Q(x)of degree≤1+3n,there holds
3 q-Harmonic number identities
Some interesting identities can be obtained by choosing different Q(x)in identity(5).We will display some examples of this class of q-harmonic number identities in this section.
The list can be endless.However,we are not bothered to extend it further.The interested reader can do that for enjoyment.
[1]Weideman J A C.Padéapproximations to the logarithm I:derivation via differential equations[J].Quaestiones Mathematicae,2005,28(3):375-390.
[2]Driver K,Prodinger H,Schneider C,et al.Padéapproximations to the logarithmⅡ:identities,recurrences,and symbolic computation[J].Ramanujan Journal,2006,11(2):139-158.
[3]Driver K,Prodinger H,Schneider C,et al.Padéapproximations to the logarithm Ⅲ:alternative methods and additional results[J].Ramanujan Journal,2006,12(3):299-314.
[4]Chu Wenchang.Partial-fraction decompositions and harmonic number identities[J].Journal of Combinatorial Mathematics and Combinatorial Computating,2007,60:139-153.
[5]Chu Wenchang,Fu Mei.Dougall-Dixon formula and harmonic number identities[J].Ramanujan Journal,2009,18(1):11-31.
Weideman公式的一种q-模拟
郑德印,陈 广
(杭州师范大学理学院,浙江 杭州 310036)
使用部分分式方法将一类有理函数分解为部分分式,进而建立了一个一般化的q-harmonic数恒等式.作为例子,列出了此恒等式的12种特殊情况,得到了12个漂亮的类q-Weideman公式.
q-二项式系数;q-harmonic数;代数恒等式
date:2010-06-24
Supported by the Natural Science Foundation of Zhejiang Province of China(Y7080320).
Biography:ZHENG De-yin(1964—),male,born in Tongbai,Henan Province,associate professor,engaged in combinatorics,hypergeometric series and special function.E-mail:deyinzheng@yahoo.com.cn
O157.1 MSC2010:05A30;11B65Article character:A
1674-232X(2011)01-0011-04
10.3969/j.issn.1674-232X.2011.01.002