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I am trying to somewhat simplify a series, whose coefficients feature generalised hypergeometric functions ${}_1F_2(1;a,a+1;z)$. I was unable to find useful functional relations for this specific combination of parameters (I tried the new NIST Handbook, third volume of A.P. Prudnikov, Brychkov & Marychev's "Integrals and Series" and all online sources I could get my hands on).

Interestingly, L.J. Slater mentions on p.47 of her book "Generalized Hypergeometric Functions" that

${}_1F_2(a;b,c;z)$ is the product of two confluent hypergeometric functions

and gives a reference to the paper of F.J.W. Whipple (1927) in J. Lond. Math. Soc., 2, p. 85, which is focussed on relationships between functions ${}_3F_2$ and ${}_4F_3$. I must be missing something here, but I cannot figure out how Whipple's paper supports the Slater's statement. Therefore, my question is

  1. Is it really possible to represent generalized hypergeometric functions ${}_1F_2(a;b,c;z)$ with arbitrary (within reason) parameters $a$, $b$ and $c$ as a product of two confluent hypergeometric functions?
  2. If yes, could you please point me in a direction of the relevant book/paper/formula/derivation?
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  • $\begingroup$ This is a proper link to the mentioned paper: F.J.W. Whipple (1927) "", J. Lond. Math. Soc., **2**(2), pp. 85-90. $\endgroup$ Jun 3, 2012 at 0:26
  • $\begingroup$ Maple expresses your function in terms of a Lommel S1 function: $${\mbox{$_1$F$_2$}(1;\,a,a+1;\,z)}=4\,{\frac {{\it LommelS1} \left( -2+ 2\,a,1,2\,\sqrt {-z} \right) \left( a-1 \right) a}{ \left( 2\,\sqrt { -z} \right) ^{2\,a-1}}} $$ $\endgroup$ Jun 3, 2012 at 6:26
  • $\begingroup$ @Robert Israel: This is very interesting. Could you please explain how you obtained this identity? I use Maple 13 and have not been able to obtain something this compact directly. $\endgroup$ Jun 3, 2012 at 8:12
  • $\begingroup$ Just found a formula underlying Robert Israel's result in Watson's "Treatise on the Theory of Bessel functions" (1966), Section 10.7, Equation (10): \begin{align} & s_{\mu,\nu}(z)=\frac{z^{\mu+1}}{(\mu-\nu+1)(\mu+\nu+1)} \\ &\qquad\qquad \times{}_1F_2\left( {\textstyle 1;\frac{1}{2}\mu-\frac{1}{2}\nu+\frac{3}{2},\frac{1}{2}\mu+\frac{1}{2}\nu+\frac{3}{2};-\frac{1}{4}z^2} \right)\,. \end{align} $\endgroup$ Jun 3, 2012 at 8:47
  • $\begingroup$ In Maple 16: > FunctionAdvisor(specialize, hypergeom([1],[a,a+1],z)); $\endgroup$ Jun 4, 2012 at 5:42

1 Answer 1

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It seems the relevant paper is in fact C. T. Preece (1923) in Proceedings of The London Mathematical Society, Volume s2-22, Issue 1, p. 370. Identities such as

$ \left\{ {}_1\mathcal{F}_1\left(a;2a;x\right)\right\}^2 = e^x {}_1\mathcal{F}_2(a;2a,a+1/2;x^2/4) $

are provided. There don't seem to be any with more than one free parameter, however.

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