scholarly journals On the additive formulae of the theta functions and a collection of Lambert series pertaining to the modular equations of degree $5$

1994 ◽  
Vol 345 (1) ◽  
pp. 323-345 ◽  
Author(s):  
Li-Chien Shen
Keyword(s):  
Theta Functions ◽  
Modular Equations ◽  
Lambert Series

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 A simple proof of an expansion of an eta-quotient as a Lambert series

2005 ◽  
Vol 71 (3) ◽  
pp. 353-358 ◽  
Author(s):  
Shaun Cooper
Keyword(s):  
Simple Proof ◽  
Theta Functions ◽  
Definite Integral ◽  
Lambert Series

We give a simple proof of the identity The proof uses only a few well-known properties of the cubic theta functions a(q), b(q) and c(q). We show this identity implies the interesting definite integral .

scholarly journals On a pair of Ramanujan’s modular equations and P - Q theta functions of level 35

2019 ◽  
Vol 43 (1) ◽  
pp. 63-80
Author(s):  
Kaliyur Ranganna VASUKI ◽  
Anusha THIPPESHA
Keyword(s):  
Theta Functions ◽  
Modular Equations

ON THE SCHRÖTER FORMULA FOR THETA FUNCTIONS

2009 ◽  
Vol 05 (08) ◽  
pp. 1477-1488 ◽  
Author(s):  
ZHI-GUO LIU ◽  
XIAO-MEI YANG
Keyword(s):  
Theta Function ◽  
Theta Functions ◽  
Modular Equations ◽  
Trigonometric Identity ◽  
Product Identity ◽  
Function Identity ◽  
Special Cases ◽  
Theta Function Identity ◽  
Addition Formulas

The Schröter formula is an important theta function identity. In this paper, we will point out that some well-known addition formulas for theta functions are special cases of the Schröter formula. We further show that the Hirschhorn septuple product identity can also be derived from this formula. In addition, this formula allows us to derive four remarkable theta functions identities, two of them are extensions of two well-known Ramanujan's identities related to the modular equations of degree 5. A trigonometric identity is also proved.

scholarly journals A Note on Certain Modular Equations about Infinite Products of Ramanujan

2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Hong-Cun Zhai
Keyword(s):  
Theta Functions ◽  
Modular Equations ◽  
Infinite Products

Ramanujan proposed additive formulae of theta functions that are related to modular equations about infinite products. Employing these formulaes, we derived some identities on infinite products. In the same spirit, we also could present elementary and simple proofs of certain Ramanujan's modular equations on infinite products.

The H and K Family of Mock Theta Functions

2012 ◽  
Vol 64 (4) ◽  
pp. 935-960 ◽  
Author(s):  
Richard J. McIntosh
Keyword(s):  
Rational Number ◽  
Theta Functions ◽  
Mock Theta Functions ◽  
Lambert Series ◽  
Linear Relations ◽  
Root Of Unity ◽  
Lost Notebook ◽  
Modular Transformations ◽  
Ramanujan's Lost Notebook ◽  
Even Order

AbstractIn his last letter to Hardy, Ramanujan defined 17 functionsF(q), |q| < 1, which he calledmockθ-functions. He observed that asqradially approaches any root of unity ζ at whichF(q) has an exponential singularity, there is aθ-functionTζ(q) withF(q) −Tζ(q) =O(1). Since then, other functions have been found that possess this property. These functions are related to a functionH(x,q), wherexis usuallyqrore2πirfor some rational numberr. For this reason we refer toHas a “universal” mockθ-function. Modular transformations ofHgive rise to the functionsK,K1,K2. The functionsKandK1appear in Ramanujan's lost notebook. We prove various linear relations between these functions using Appell–Lerch sums (also called generalized Lambert series). Some relations (mock theta “conjectures”) involving mockθ-functions of even order andHare listed.

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