Expectation of a Fraction of Gaussian Hypergeometric Functions

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Discussion Overview

The discussion revolves around the expectation of a fraction of Gauss hypergeometric functions, specifically seeking identities or methods to simplify or express this fraction. The scope includes theoretical aspects of hypergeometric functions and their applications in probability or statistics.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant seeks to find the expectation of a fraction of Gauss hypergeometric functions and asks for identities that could simplify the expression.
  • Another participant questions the definition of the random variable involved in the expectation, seeking clarification on what variable is being referenced.
  • A later reply confirms that the random variable is x and reiterates the request for ideas on simplifying the fraction.
  • One participant admits to having no experience with hypergeometric functions and cannot provide assistance.
  • Another participant suggests that there are many identities related to hypergeometric functions and references specific literature, including works by Erdélyi and the NIST handbook, as potential resources for relevant identities.
  • A participant provides a summary of the original question and attempts to clarify the notation used for hypergeometric functions, comparing it to more familiar forms and suggesting that the notation may indicate a generalized hypergeometric series.
  • There is a mention of the NIST handbook as a modern reference for hypergeometric identities, indicating a shift in available resources.

Areas of Agreement / Disagreement

Participants express varying levels of familiarity with hypergeometric functions, leading to some uncertainty about the notation and the specific identities that may apply. There is no consensus on how to proceed with the original question, and multiple viewpoints on the resources and identities available remain present.

Contextual Notes

There are limitations regarding the clarity of notation used for hypergeometric functions, and the discussion does not resolve the mathematical steps necessary to simplify the expectation expression.

rafgger
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TL;DR
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!
 
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The term "expectation" usually refers to a random variable. What is the variable here?
 
mathman said:
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.
 
Sorry I have no idea. I have never worked with these functions.
 
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.
 
rafgger said:
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.
 

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