A general q -expansion formula based on matrix inversions and its applications
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A general q-expansion formula based on matrix inversions and its applications Jin Wang1 Received: 22 May 2018 / Accepted: 8 August 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this paper, by the technique of matrix inversions, we establish a general q-expansion formula of arbitrary formal power series F(z) with respect to the base
n = 0, 1, 2, . . . . z (bz)n n (az)n
Some concrete expansion formulas and their applications to q-series identities are presented, including Carlitz’s q-expansion formula and a new partial theta function identity as well as a coefficient identity for Ramanujan’s 1 ψ1 summation formula as special cases. Keywords Matrix inversion · Expansion formula · Coefficient · q-Series · Identity · Lagrange–Bürmann inversion · Formal power series · q-Catalan number Mathematics Subject Classification Primary 05A30 · Secondary 33D15
1 Introduction Throughout the present paper, we adopt the standard notation and terminology for q-series from [7] due to Gasper and Rahman. The q-shifted factorials of complex
This work was supported by the Natural Science Foundation of Zhejiang Province (Grant No. LQ20A010004) and by the National Natural Science Foundation of China (Grant Nos. 11471237 and 11971341).
B 1
Jin Wang [email protected] Department of Mathematics, Zhejiang Normal University, Jinhua 321004, People’s Republic of China
123
J. Wang
variable z with the base q : |q| < 1 are given by (z)∞ := (z; q)∞ =
∞
(1 − zq n ) and
n=0
(z)∞ (z)n := (zq n )∞
(1.1)
for all integers n. For integer m ≥ 1, we use the notation (a1 , a2 , . . . , am )n := (a1 )n (a2 )n . . . (am )n . Also, the r +1 φr series with the base q and the argument z is defined to be r +1 φr
∞ a1 , a2 , . . . , ar +1 (a1 , a2 , . . . , ar +1 )n n ; q, z := z . b1 , b2 , . . . , br (q, b1 , b2 , . . . , br )n n=0
For any f (z) = n≥0 an z n ∈ C[[z]], where C[[z]] denotes the ring of formal power series in variable z, we shall employ the coefficient functional [z n ]{ f (z)} := an and a0 = f (0). We also follow the summation convention that for any integers m and n, n k=m
ak = −
m−1
ak .
k=n+1
Recall that in [5], Coogan and Ono showed ∞ n=0
∞
zn
(z)n 2 = (−1)n z 2n q n (−z)n+1
(1.2)
n=0
and used it to find the generating functions for values of certain expressions of the Hurwitz zeta function at non-positive integers. The appearance of (1.2) reminds us of the famous Rogers–Fine identity [11, Eq. (17.6.12)]: (1 − z)
∞ n=0
∞
zn
(aq)n 2 (aq, azq/b)n = (1 − azq 2n+1 )(bz)n q n . (bq)n (bq, zq)n
(1.3)
n=0
As a matter of fact, (1.2) can be easily deduced from (1.3) by letting aq = z = −b. Moreover, by letting a = z = −b in (1.3), we obtain another identity as follows: ∞ n=0
123
∞
zn
(z)n+1 2 =1+2 (−1)n z 2n q n . (−zq)n n=1
(1.4)
A general q-expansion formula based on matrix...
It is these identities, once treated as formal power series in z, that make us be aware of investigating in a full generality the problem of representations o
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