Expectation of a Fraction of Gaussian Hypergeometric Functions

[rafgger]

TL;DR Summary

In this question I am seeking an expression for the fraction of particular hypergeometric functions and more, their expectation.

I am looking for the expectation of a fraction of Gauss hypergeometric functions.

$$E_X\left[\frac{{}_2F_1\left(\begin{matrix}x+a+1\\x+a+1\end{matrix},a+1,c\right)}{{}_2F_1\left(\begin{matrix}x+a\\x+a\end{matrix},a,c\right)}\right]=?$$

Are there any identities that could be used to simplify or express the fraction?Or wouldn't an idea, how to proceed?

Thank you very much!

 

[mathman]

The term "expectation" usually refers to a random variable. What is the variable here?

 

[rafgger]

The term "expectation" usually refers to a random variable. What is the variable here?

The random variable is x. Even, if there would be an idea, how to simplify the fraction. Would be most appreciated.

 

[mathman]

Sorry I have no idea. I have never worked with these functions.

 

[jim mcnamara]

AFAIK there are many identities and the one you may think relevant is probably in:

see: Erdélyi, Arthur; Magnus, Wilhelm; Oberhettinger, Fritz & Tricomi, Francesco G. (1953). Higher transcendental functions (PDF). Vol. I. New York – Toronto – London: McGraw–Hill Book Company, Inc. ISBN 978-0-89874-206-0. MR 0058756.

Or the NIST handbook:
Olde Daalhuis, Adri B. (2010), "Hypergeometric function", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248

I can't be of much help, possibly @Stephen Tashi may know more.

 

[TeethWhitener]

Summary:: In this question I am seeking an expression for the fraction of particular hypergeometric functions and more, their expectation.

I am looking for the expectation of a fraction of Gauss hypergeometric functions.

$$E_X\left[\frac{{}_2F_1\left(\begin{matrix}x+a+1\\x+a+1\end{matrix},a+1,c\right)}{{}_2F_1\left(\begin{matrix}x+a\\x+a\end{matrix},a,c\right)}\right]=?$$

Are there any identities that could be used to simplify or express the fraction?Or wouldn't an idea, how to proceed?

Thank you very much!

I'm not sure how to parse the notation. Typically, I've seen
$$_2F_1(a,b;c;x)=\sum_{k=0}^\infty \frac{(a)_k(b)_k}{(c)_k}\frac{x^k}{k!}$$
where $(a)_k$ denotes the Pochhammer symbol. The stacked notation you use is more indicative of a generalized hypergeometric series:
$$_2F_1\left(\begin{matrix}a_1\text{ } a_2\\b_1\end{matrix};x\right) =\sum_{k=0}^\infty \frac{(a_1)_k(a_2)_k}{(b_1)_k}\frac{x^k}{k!} $$
(which is the same as $_2F_1(a_1,a_2;b;x)$). But I can't quite figure out your notation. In any case, @jim mcnamara is right. Erdelyi is a good source for hypergeometric identities; I'd add Abramowitz and Stegun to that list.

EDIT: apparently Abramowitz and Stegun has entered the digital age and is now the NIST handbook that @jim mcnamara referred to in his post.