How to decrease the levels of modular integrals

1998 ◽  
Vol 41 (5) ◽  
pp. 491-497
Author(s):  
Xueli Wang ◽  
Dingyi Pei
Keyword(s):  
Modular Integrals

scholarly journals Some entries in Ramanujan's notebooks

2008 ◽  
Vol 144 (2) ◽  
pp. 255-266 ◽  
Author(s):  
W. DUKE
Keyword(s):  
Eisenstein Series ◽  
Hypergeometric Functions ◽  
Similar Nature ◽  
Negative Weight ◽  
Modular Integrals

Some of Ramanujan's original discoveries about hypergeometric functions and their relation to modular integrals, especially Eisenstein series ofnegative weight, are still not very well understood. These discoveries take the form of identities that he recorded, without proof, as entries in his notebooks. In the following sections I will introduce some of these entries, discuss their status, give new proofs of several of them and also provide new results of a similar nature.

scholarly journals SIGN CHANGES OF FOURIER COEFFICIENTS OF ENTIRE MODULAR INTEGRALS

2012 ◽  
Vol 54 (2) ◽  
pp. 355-358
Author(s):  
YOUNGJU CHOIE ◽  
WINFRIED KOHNEN
Keyword(s):  
Cusp Form ◽  
Fourier Coefficients ◽  
Congruence Subgroup ◽  
Short Note ◽  
Finitely Generated ◽  
Positive Real ◽  
Sign Changes ◽  
Multiplier System ◽  
Integral Weight ◽  
Modular Integrals

AbstractLet f be a non-zero cusp form with real Fourier coefficients a(n) (n ≥ 1) of positive real weight k and a unitary multiplier system v on a subgroup Γ ⊂ SL2(ℝ) that is finitely generated and of Fuchsian type of the first kind. Then, it is known that the sequence (a(n))(n ≥ 1) has infinitely many sign changes. In this short note, we generalise the above result to the case of entire modular integrals of non-positive integral weight k on the group Γ0*(N) (N ∈ ℕ) generated by the Hecke congruence subgroup Γ0(N) and the Fricke involution $W_N:= \big(\scriptsize\begin{array}{c@{}c} 0 & -{1/\sqrt N} \\[3pt] \sqrt N & 0\\ \end{array}\big)$ provided that the associated period functions are polynomials.

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