scholarly journals On Modular Forms Arising from a Differential Equation of Hypergeometric Type

2003 ◽  
pp. 145-164 ◽  
Author(s):  
Masanobu Kaneko ◽  
Masao Koike
Keyword(s):  
Differential Equation ◽  
Modular Forms ◽  
Hypergeometric Type

scholarly journals ON THE MODULARITY OF SOLUTIONS TO CERTAIN DIFFERENTIAL EQUATIONS OF HYPERGEOMETRIC TYPE

2021 ◽  
pp. 1-7
Author(s):  
HICHAM SABER ◽  
ABDELLAH SEBBAR
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Modular Forms ◽  
Hypergeometric Type

Abstract We answer some questions in a paper by Kaneko and Koike [‘On modular forms arising from a differential equation of hypergeometric type’, Ramanujan J.7(1–3) (2003), 145–164] about the modularity of the solutions of a certain differential equation. In particular, we provide a number-theoretic explanation of why the modularity of the solutions occurs in some cases and does not occur in others. This also proves their conjecture on the completeness of the list of modular solutions after adding some missing cases.

scholarly journals Hypergeometric type identities in the $p$-adic setting and modular forms

2015 ◽  
Vol 144 (4) ◽  
pp. 1493-1508 ◽  
Author(s):  
Jenny G. Fuselier ◽  
Dermot McCarthy
Keyword(s):  
Modular Forms ◽  
Hypergeometric Type

scholarly journals Siegel theta series for indefinite quadratic forms

2021 ◽  
Vol 7 (3) ◽  
Author(s):  
Christina Roehrig
Keyword(s):  
Differential Equation ◽  
Modular Forms ◽  
Quadratic Forms ◽  
Theta Series ◽  
Second Order ◽  
Transformation Behavior ◽  
Modular Transformation ◽  
Indefinite Quadratic

AbstractThe modular transformation behavior of theta series for indefinite quadratic forms is well understood in the case of elliptic modular forms due to Vignéras, who deduced that solving a differential equation of second order serves as a criterion for modularity. In this paper, we will give a generalization of this result to Siegel theta series.

On the asymptotic expansions of solutions of an nth order linear differential equation

1980 ◽  
Vol 85 (1-2) ◽  
pp. 15-57 ◽  
Author(s):  
R. B. Paris
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Asymptotic Expansions ◽  
Asymptotic Theory ◽  
Arbitrary Order ◽  
Order Linear Differential Equation ◽  
Linear Ordinary Differential Equations ◽  
Hypergeometric Type ◽  
Independent Variable ◽  
Linear Differential

SynopsisThe asymptotic expansions of solutions of a class of linear ordinary differential equations of arbitrary order n are investigated for large values of the independent variable z in the complex plane. Solutions are expressed in terms of Mellin-Barnes integrals and their asymptotic expansions are subsequently determined by means of the asymptotic theory of integral functions of the hypergeometric type. Three classes of solutions are considered: (i) solutions whose behaviour is either exponentially large or algebraic for |z|→∞ in different sectors of the z-plane, (ii) solutions which are even and odd functions of z when the order n of the differential equation is even and (iii) solutions which are exponentially damped as |z|→∞ in a certain sector of the z-plane.

scholarly journals Some Extensions of Pincherle's Polynomials

1920 ◽  
Vol 39 ◽  
pp. 21-24 ◽  
Author(s):  
Pierre Humbert
Keyword(s):  
Differential Equation ◽  
General Theory ◽  
Differential Equations ◽  
Second Order ◽  
Linear Differential Equations ◽  
Third Order ◽  
The Third ◽  
Hypergeometric Type ◽  
Linear Differential ◽  
Image Position

The polynomials which satisfy linear differential equations of the second order and of the hypergeometric type have been the object of extensive work, and very few properties of them remain now hidden; the student who seeks in that direction a subject for research is compelled to look, not after these functions themselves but after generalisations of them. Among these may be set in first place the polynomials connected with a differential equation of the third order and of the extended hypergeometric type, of which a general theory has been given by Goursat. The number of such polynomials of which properties have been studied in particular is rather small; in fact, Appell's polynomialsand Pincherle's polynomials, arising from the expansionsare, so far as I know, the only well-known ones. To show what can be done in these ways, I shall briefly give the definition and principal properties of some polynomials analogous to Pincherle's and of some allied functions.

scholarly journals On the Continued Fractions associated with the Hypergeometric Equation

1915 ◽  
Vol 34 ◽  
pp. 146-154 ◽  
Author(s):  
E. Lindsay Ince
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Continued Fractions ◽  
Present Author ◽  
Continued Fraction ◽  
Second Order ◽  
Hypergeometric Equation ◽  
Hypergeometric Type ◽  
Linear Differential

The present paper is based on a method attributed to Euler of expressing as a continued fraction the logarithmic derivate of a solu tion of a linear differential equation of the second order. The method is particularly applicable to equations of hypergeometric type, and, in that connection, was previously employed by the present author as a means of adding to the number of known transformations of continued fractions.

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