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How to calculate degenerate hypergeometric function of two variables

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Ronald 2017-5-8

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评论： David Goodmanson 2017-5-9

How to calculate a degenerate hypergeometric function of two variables? There is no such built-in function in matlab.

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Walter Roberson 2017-5-8

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Are those related to Confluent Hypergeometric Functions ? https://en.wikipedia.org/wiki/Confluent_hypergeometric_function

Ronald 2017-5-8

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$\Phi(\alpha,\beta,\gamma; x,y )={\Gamma(\gamma)\over \Gamma(\alpha)\Gamma(\gamma-\alpha)}\int_0^1 u^{\alpha-1}u^{\gamma-\alpha-1}(1-ux)^{-\beta}e^{uy}du

If you put the formula in the latex, you can see the form. I am afraid if the matlab allows latex type-in feature. It would be great if you could help me out. \Gamma is a gamma function

Walter Roberson 2017-5-8

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Walter Roberson 2017-5-8

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That is the same formula you posted before.

I was not able to find any useful transformation to other functions.

I was not able to find information about "degenerate hypergeometric function" under that name; is there a different name for it?

David Goodmanson 2017-5-8

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编辑：David Goodmanson 2017-5-8

It is indeed called the confluent hypergeometric function in two variables, denoted by phi_1, only the factor u^(g-a-1) above should be (1-u)^(g-a-1). See 'Humbert series' in Wikipedia.

Maybe someday Mathworks will come out with a really good Mathematical Functions toolbox to replace the ad hoc collection of stuff they have now (including user contributions in the File Exchange). It doesn't seem likely any time soon though, because customer demand is taking the toolbox monster in other directions.

Ronald 2017-5-8

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Thank you for your correction, David. Hey Walter, it is called the confluent hypergeometric function in two variables. Is it possible to solve it via the command of quad or integral? If so, what should I be aware of before I use the commands?

Walter Roberson 2017-5-8

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An old post pointed to http://www.mathworks.com/matlabcentral/fileexchange/6218-computation-of-special-functions . Also perhaps see https://www.math.ucla.edu/~mason/research/pearson_final.pdf

David Goodmanson 2017-5-9

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