Generalized Lambert Series and Euler’s Pentagonal Number Theorem

2021 ◽  
Vol 18 (1) ◽  
Author(s):  
Mircea Merca
Keyword(s):  
Lambert Series ◽  
Pentagonal Number Theorem

scholarly journals Asymptotic expansions of Lambert series and related q-series

2017 ◽  
Vol 13 (08) ◽  
pp. 2097-2113 ◽  
Author(s):  
Shubho Banerjee ◽  
Blake Wilkerson
Keyword(s):  
Asymptotic Expansion ◽  
Asymptotic Expansions ◽  
Theta Functions ◽  
Lambert Series ◽  
Pochhammer Symbol ◽  
Complete Asymptotic Expansion ◽  
Polygamma Functions ◽  
Jacobi Theta Functions ◽  
Relative Errors ◽  
Asymptotic Forms

We study the Lambert series [Formula: see text], for all [Formula: see text]. We obtain the complete asymptotic expansion of [Formula: see text] near [Formula: see text]. Our analysis of the Lambert series yields the asymptotic forms for several related [Formula: see text]-series: the [Formula: see text]-gamma and [Formula: see text]-polygamma functions, the [Formula: see text]-Pochhammer symbol and the Jacobi theta functions. Some typical results include [Formula: see text] and [Formula: see text], with relative errors of order [Formula: see text] and [Formula: see text] respectively.

scholarly journals Linear independence of certain Lambert series

2014 ◽  
Vol 142 (10) ◽  
pp. 3411-3419 ◽  
Author(s):  
Florian Luca ◽  
Yohei Tachiya
Keyword(s):  
Linear Independence ◽  
Lambert Series

Generalized Lambert series

1996 ◽  
pp. 357-370 ◽  
Author(s):  
Ronald Evans
Keyword(s):  
Lambert Series

scholarly journals Truncated Series with Nonnegative Coefficients from the Jacobi Triple Product

2022 ◽  
Vol Volume 44 - Special... ◽  
Author(s):  
Liuquan Wang
Keyword(s):  
Analogous Result ◽  
Triple Product ◽  
Combinatorial Proof ◽  
Jacobi’S Triple Product Identity ◽  
Product Identity ◽  
Pentagonal Number Theorem ◽  
Nonnegative Coefficients ◽  
Jacobi's Triple Product Identity

Andrews and Merca investigated a truncated version of Euler's pentagonal number theorem and showed that the coefficients of the truncated series are nonnegative. They also considered the truncated series arising from Jacobi's triple product identity, and they conjectured that its coefficients are nonnegative. This conjecture was posed by Guo and Zeng independently and confirmed by Mao and Yee using different approaches. In this paper, we provide a new combinatorial proof of their nonnegativity result related to Euler's pentagonal number theorem. Meanwhile, we find an analogous result for a truncated series arising from Jacobi's triple product identity in a different manner.

scholarly journals Bijections and Congruences for Generalizations of Partition Identities of Euler and Guy

10.37236/1796 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
James A. Sellers ◽  
Andrew V. Sills ◽  
Gary L. Mullen
Keyword(s):  
Famous Theorem ◽  
Partition Identities ◽  
Product Identity ◽  
Powers Of 2 ◽  
Pentagonal Number Theorem ◽  
Series Product

In 1958, Richard Guy proved that the number of partitions of $n$ into odd parts greater than one equals the number of partitions of $n$ into distinct parts with no powers of 2 allowed, which is closely related to Euler's famous theorem that the number of partitions of $n$ into odd parts equals the number of partitions of $n$ into distinct parts. We consider extensions of Guy's result, which naturally lead to a new algorithm for producing bijections between various equivalent partition ideals of order 1, as well as to two new infinite families of parity results which follow from Euler's Pentagonal Number Theorem and a well-known series-product identity of Jacobi.

Lambert series associated to Hilbert modular form

2021 ◽  
pp. 1-15
Author(s):  
Rishabh Agnihotri
Keyword(s):  
Asymptotic Expansion ◽  
Modular Form ◽  
Cusp Form ◽  
Riemann Zeta Function ◽  
Hilbert Modular Form ◽  
Lambert Series ◽  
Nontrivial Zeros ◽  
Hilbert Modular ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

In 1981, Zagier conjectured that the Lambert series associated to the weight 12 cusp form [Formula: see text] should have an asymptotic expansion in terms of the nontrivial zeros of the Riemann zeta function. This conjecture was proven by Hafner and Stopple. In 2017 and 2019, Chakraborty et al. established an asymptotic relation between Lambert series associated to any primitive cusp form (for full modular group, congruence subgroup and in Maass form case) and the nontrivial zeros of the Riemann zeta function. In this paper, we study Lambert series associated with primitive Hilbert modular form and establish similar kind of asymptotic expansion.

On the sum of parts with multiplicity at least 2 in all the partitions of n

2020 ◽  
pp. 1-17
Author(s):  
Mircea Merca ◽  
Ae Ja Yee
Keyword(s):  
Combinatorial Proof ◽  
Combinatorial Interpretations ◽  
Pentagonal Number Theorem

In this paper, we investigate the sum of distinct parts that appear at least 2 times in all the partitions of [Formula: see text] providing new combinatorial interpretations for this sum. A connection with subsets of [Formula: see text] is given in this context. We provide two different proofs of our results: analytic and combinatorial. In addition, considering the multiplicity of parts in a partition, we provide a combinatorial proof of the truncated pentagonal number theorem of Andrews and Merca.

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