I am looking for a reference for the two following equalities involving the Tricomi function $U$ and the Meijer function $G$. I have found these formulas on the website http://functions.wolfram.com/, is it possible to refer to it in an article (and how) ? $$ \int_0^y x^{a-1}U(\alpha, \beta, x) d x = \frac{1}{\Gamma(\alpha)\Gamma(\alpha-\beta+1)} G_{2,3}^{2,2}\left( y \Bigm| \begin{matrix} 1, a-\alpha+1 \\ a, a-\beta+1, 0 \end{matrix}\right). $$ $$ \int_0^y \exp(-x)x^{a-1}U(\alpha, \beta, x) d x = G_{2,3}^{2,1}\left( y \Bigm| \begin{matrix} 1, a+\alpha-\beta+1 \\ a, a-\beta+1, 0 \end{matrix}\right). $$
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2$\begingroup$ On Wolfram's function site the formula I found for your second integral has a $G_{2,3}^{2,1}$ on the r.h.s. This would also be consistent with the number of upper and lower parameters. Maybe you want to correct that. $\endgroup$– Johannes TrostCommented Jul 4, 2015 at 21:57
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4 Answers
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indefinite integrals of the type $$\int x^pe^{qx}U(\alpha, \beta, x) d x$$ were considered in http://cdm16009.contentdm.oclc.org/cdm/ref/collection/p13011coll6/id/61450 (On some indefinite integrals of confluent hypergeometric functions, by E.W. Ng and M. Geller). However their results are not expressed through the Meijer function.
answered Jul 6, 2015 at 5:33
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$\begingroup$ Thank you for the source. However, it seems that it no longer works. Can you please share an alternate source? $\endgroup$ Commented Apr 14, 2018 at 10:38
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$\begingroup$ ia800501.us.archive.org/7/items/jresv74Bn2p85/jresv74Bn2p85.pdf - An Alternate source $\endgroup$ Commented Apr 14, 2018 at 11:10
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$\begingroup$ Here is the link for where the source is still available ia800501.us.archive.org/7/items/jresv74Bn2p85/jresv74Bn2p85.pdf $\endgroup$ Commented Apr 14, 2018 at 18:22
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Many of the identities in functions.wolfram.com were semi-automatically generated and checked from the simplification and transformation rules known by the Mathematica Kernel as well as tested by injection of values for the variable and parameters, etc. So they may never have been printed in traditional books or journals.
Here is the link to the description of permanent urls to the functions.wolfram.com site
How to Cite Identities and Formulas from the Mathematical Functions Website
that you can use to refer not only to a particular formula but to the specific version you used (in the case it is corrected afterhand on the website).
For instance the first one you quote looks like
http://functions.wolfram.com/07.33.21.0003.01
And it may be possible to rederive it by yourself from the representation of U as a Meijer function
http://functions.wolfram.com/07.33.26.0004.01
$$U(a,b,z) = \frac{1}{\Gamma (a) \Gamma (a-b+1)}G_{1,2}^{2,1}\left(z\left| \begin{array}{c} 1-a \\ 0,1-b \\ \end{array} \right.\right)$$
and the powerful transform formula
http://functions.wolfram.com/07.34.21.0084.01
(I prefer not to reproduce here not to mangle its content)
as it may be close to the way it was initially derived.
I do not have handy here the five volumes of "Integrals and Series" (Prudnikov, Brychkov, Marichev), but your formulas may very well be in them as well. Oleg Marichev is one of the main contributors of functions.wolfram.com.
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$\begingroup$ Thank you. But what would be the complete reference ? Something like Wolfram: functions.wolfram.com/07.33.21.0003.01 ? $\endgroup$ Commented Jul 4, 2015 at 15:22
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$\begingroup$ Perhaps something like O. Marichev, M. Trott, and S. Wolfram, formula functions.wolfram.com/07.33.21.0003.01 . It depends on the quoting style of the venue you are publishing in. $\endgroup$– ogerardCommented Jul 5, 2015 at 8:17
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$\begingroup$ Thank you for your help. There are some examples in the references of this paper. brucehardie.com/notes/020/… $\endgroup$ Commented Jul 6, 2015 at 12:06
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$\begingroup$ Bibtex entries proposed here: tex.stackexchange.com/questions/51741/… Using Wolfram Research, Inc. as author and publisher sounds good, as here: support.wolfram.com/kb/472 $\endgroup$ Commented Jul 6, 2015 at 17:36
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Inspired by the links I posted in some comments to @ogerard's answer, I propose this Bibtex entry for citing a formula of the Wolfram functions site:
@online{tricomi1,
author = {Wolfram Research, Inc.},
publisher = {The Wolfram Functions Site},
title = {Tricomi confluent hypergeometric function},
subtitle = {Integration (formula 07.33.21.0003)},
note = {Visited on 06/07/2015},
url = {http://functions.wolfram.com/07.33.21.0003.01}
}
The subtitle is the title of the page http://functions.wolfram.com/07.33.21.0003.01 To get it, I open the html source code of the page and it is in the <title> tag.
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Another thing to try: The Meijer G-function is solution of a certain differential equation. Maybe verify that the left-hand-side satisfies that differential equation.
answered Jul 4, 2015 at 14:21
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$\begingroup$ Yes, but I am not looking for the proofs of the formulas, only references. $\endgroup$ Commented Jul 4, 2015 at 15:23
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2$\begingroup$ I checked Gradshteyn & Ryzhik and did not find them. $\endgroup$ Commented Jul 4, 2015 at 15:26