Can one represent a generalized hypergeometric function 1F2 as a product of two confluent hypergeometric functions? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T15:42:00Z http://mathoverflow.net/feeds/question/98684 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/98684/can-one-represent-a-generalized-hypergeometric-function-1f2-as-a-product-of-two-c Can one represent a generalized hypergeometric function 1F2 as a product of two confluent hypergeometric functions? Aleksey Pichugin 2012-06-02T21:43:13Z 2012-06-02T22:50:41Z <p>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 &amp; Marychev's "Integrals and Series" and all online sources I could get my hands on).</p> <p>Interestingly, L.J. Slater mentions on p.47 of her book "Generalized Hypergeometric Functions" that</p> <p>\${}_1F_2(a;b,c;z)\$ <em>is the product of two confluent hypergeometric functions</em></p> <p>and gives a reference to the paper of F.J.W. Whipple (1927) in <em>J. Lond. Math. Soc.</em>, 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</p> <ol> <li>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?</li> <li>If yes, could you please point me in a direction of the relevant book/paper/formula/derivation?</li> </ol>