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According to wolfram, A generalization of the confluent hypergeometric differential equation is given by;

$$y''+\left(\frac{2R}{x}+2F'+p\frac{H'}{H}-H'-\frac{H''}{H'}\right) y'+\left[\left(p\frac{H'}{H}-H'-\frac{H''}{H'}\right)\left(\frac{R}{x}+F'\right)+\frac{R(R-1)}{x^2}+\frac{2R}{x}F'+F''+(F')^2-\frac{q}{H}(H')^2 \right]y=0$$

Which has the solutions $y_1=x^{-R} e^{-F} M(q,p,H)$ and $y_2=x^{-R} e^{-F} O(q,p,H)$, where $M(q,p,H)$ is the confluent hypergeometric function of the first kind and $O(q,p,H)$ is the confluent hypergeometric function of the second kind. Meanwhile, $R,F$ and $H$ are fucntions of $x$.

I tried to look on google for more details about this equation but i didn't find anything, can anyone here please give me more references about this particular equation? Like how it was deriven, the relation between the parameters $p$ and $q$..etc.

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    $\begingroup$ Take $X = H(x), \,y(x) = x^{-R} e^{-F(x)} Y(X)$, you'll get $$\frac {(H'(x))^2} {x^R e^{F(x)} H(x)} (X Y''(X) + (p - X) Y'(X) - q Y(X)) = 0.$$ $\endgroup$
    – Maxim
    Commented Oct 22, 2020 at 12:53
  • $\begingroup$ Can you please explain how you got your solution? @Maxim $\endgroup$ Commented Nov 11, 2020 at 17:02

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A relevant book or textbook would be the following one:

The Confluent Hypergeometric Function: with Special Emphasis on its Applications, by Herbert Buchholz. The first chapter is entitled The Various Forms of the Differential Equation for the Confluent Hypergeometric Function and the Definitions of their Solutions.

Another helpful book with historical notes is Generalized Hypergeometric Functions, by Lucy Joan Slater.

Other related books/textbooks/references include the following ones:

Generalized Hypergeometric Functions, by Bernard Dwork.

Generalized Hypergeometric Series, by W.N. Bailey.

Basic Hypergeometric Series (2nd edition), by George Gasper and Mizan Rahman.

Theory of Hypergeometric Functions, by Kazuhiko Aomoto and Michitake Kita. This book deals among other things with the geometric theory of complex analytic integrals representing hypergeometric functions of several variables, in relation to cohomology theory.

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  • $\begingroup$ Your boobs are helpful E. Noujeim, but they don't contain the equation i'm looking for. $\endgroup$ Commented Nov 9, 2020 at 20:09
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Theory of hypergeometric functions: The book presents a geometric theory of complex analytic integrals representing hypergeometric functions of several variables. Starting from an integrand which is a product of powers of polynomials, integrals are explained, in an open affine space, as a pair of twisted de Rham cohomology and its dual over the coefficients of local system. It is shown that hypergeometric integrals generally satisfy a holonomic system of linear differential equations with respect to the coefficients of polynomials and also satisfy a holonomic system of linear difference equations with respect to the exponents. These are deduced from Grothendieck-Deligne’s rational de Rham cohomology on the one hand, and by multidimensional extension of Birkhoff’s classical theory on analytic difference equations on the other.

Hypergeometric summation. An algorithmic approach to summation and special function identities: Modern algorithmic techniques for summation most of which have been introduced within the last decade are developed and carefully implemented via computer algebra system software (which can be downloaded from the Web). The algorithms of Gosper, Zeilberger, and Petkovsek on hypergeometric summation and recurrence equations and their $q$-analogues are covered, and similar algorithms on differential equations are considered. An equivalent theory of hyperexponential integration due to Almkvist and Zeilberger completes the volume. The combination of all results considered gives work with orthogonal polynomials and (hypergeometric type) special functions a solid algorithmic foundation. Hence, many examples from this very active field are given. The book is designed for use as framework for a seminar on the topic, but is also suitable for use in an advanced lecture course.

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