Normalized Fox-Wright function

Function

definition	$??$
definition	${}_p\Psi_q^*[\langle a_1, A_1\rangle,…,\langle a_p, A_p\rangle; \langle b_1, B_1\rangle,…,\langle a_q, A_q\rangle](z):= \sum_{n=0}^\infty c_n z^n$
with	$c_n = \dfrac{1}{n!}\dfrac{\prod_{m=1}^p \Gamma(a_m+A_m\cdot{n})\, /\, \Gamma(a_m)}{\prod_{j=1}^q \Gamma(b_j+B_m\cdot{n})\, /\, \Gamma(b_j)}$

Discussion

Elaboration/Motivation

The coefficients of ${}_p\Psi_q^*$ relate very similarly to each other as is the case for the Generalized hypergeometric function. The latter function is indeed obtained as Special case when all capital letters equal to $1$ .

How to read the factors

Note that for $n,m\in{\mathbb N}$ we have

$\dfrac{\Gamma(n+m+1)}{\Gamma(m+1)} = \dfrac{(n+m)!}{m!} = \dfrac{\prod_{j=1}^{n+m}j}{\prod_{i=1}^m i} = \prod_{j=m+1}^{m+n} j = (m+1)\cdot(m+2)\cdots(m+n).$

So an expansion coefficient of ${}_p\Psi_q^*$ is a fraction of products with factors $\Gamma(a_m+A_m\cdot{n})\,/\,\Gamma(a_m)$ , which are essentially also a product consisting of equidistant factors.

E.g. at $n=5$ , the context $\langle a_1,A_1\rangle=\langle 5,1\rangle$ gives a multiplicative contribution

$\dfrac{\Gamma(4+5)}{\Gamma(4)} = 4 \cdot 5\cdot 6\cdot 7\cdot 8$ .

Reference

Wikipedia: Normalized fox-wright function

Subset of

Function

Generalized hypergeometric function