# Integrals involving the Tricomi hypergeometric function

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) ? $$\int_0^y x^{a-1}U(\alpha, \beta, x) d x = \frac{1}{\Gamma(\alpha)\Gamma(\alpha-\beta+1)} G_{2,3}^{2,2}\left( y \Bigm| \begin{matrix} 1, a-\alpha+1 \\ a, a-\beta+1, 0 \end{matrix}\right).$$ $$\int_0^y \exp(-x)x^{a-1}U(\alpha, \beta, x) d x = G_{2,3}^{2,1}\left( y \Bigm| \begin{matrix} 1, a+\alpha-\beta+1 \\ a, a-\beta+1, 0 \end{matrix}\right).$$

• 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. – Johannes Trost Jul 4 '15 at 21:57

## 4 Answers

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.

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.

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.

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.

• Yes, but I am not looking for the proofs of the formulas, only references. – Stéphane Laurent Jul 4 '15 at 15:23
• I checked Gradshteyn & Ryzhik and did not find them. – Gerald Edgar Jul 4 '15 at 15:26