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

- 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?
- If yes, could you please point me in a direction of the relevant book/paper/formula/derivation?

J. Lond. Math. Soc.,**2**(2), pp. 85-90. $\endgroup$2more comments