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Hypergeometric series with gamma product formula. (English) Zbl 1382.33001

Let \(\mathbb C\) and \(D\) denote the complex plane and the unit disk in the complex plane, respectively. For a given data \(\lambda=(p,q,r;a,b;x)\in \mathbb C^5\times D\) consider an entire meromorphic function \[ f(w;\lambda):= {}_2F_1(pw+a,qw+b;rw;x), \]
where \({}_2F_1(\alpha,\beta;\gamma;z)\) is the Gauss hypergeometric series. The function \(f(w;\lambda)\) is said to admit a gamma product formula (GPF) if there exist a rational function \(S(w)\in\mathbb C(w)\); a constant \(d\in\mathbb C^\times\); two integers \(m,n\in\mathbb Z_{\geq 0}\); \(m\) numbers \(u_1,\ldots,u_m\in \mathbb C\); \(n\) numbers \(v_1,\ldots,v_n\in \mathbb C\) such that
\[ f(w;\lambda)=S(w)\cdot d^w \cdot \frac{\Gamma(w+u_1)\dots \Gamma(w+u_m)}{\Gamma(w+v_1)\dots \Gamma(w+v_n)}, \]
where \(\Gamma(w)\) is the Euler gamma function. The function \(f(w;\lambda)\) is also said to be of closed form if \[ \frac{f(w+1;\lambda)}{f(w;\lambda)}=:\mathbb R(w;\lambda)\in \mathbb C(w)\;\text{: a rational function of \(w\).} \]
The author considers two problems:
Problem I. Find a data \(\lambda=(p,q,r;a,b;x)\) for which \(f(w;\lambda)\) admits GPF; and
Problem II. Find a data \(\lambda=(p,q,r;a,b;x)\) for which \(f(w;\lambda)\) is of closed form.
The author focuses on \(\lambda\) lying in the real domain, that is,
\[ p,q,r\in \mathbb R,\quad 0< p< r\;\text{ or }\;0< q< r;\quad a,b\in \mathbb R;\quad 0< x< 1 \]
and proves that in that case Problem I and Problem II are equivalent (Thm. 2.1). Consequently, the author gives some necessary conditions (Thms. 2.2–2.5) for a given data \(\lambda\) to be a solution to Problems I and II.

MSC:

33B15 Gamma, beta and polygamma functions
33C05 Classical hypergeometric functions, \({}_2F_1\)

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References:

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