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Log-concavity for series in reciprocal gamma functions and applications. (English) Zbl 1286.26008

The authors study convexity properties of series of type \[ \sum^\infty_{n=0} f_n\cdot x^n/n!\Gamma(a+ n)= g(a,x) \] (here \(f_n)\) is a special log-concave sequence). They prove e.g. that for fixed \(x\geq 0\) \(g(a,x)\) is a strictly log-concave function of \(a\) on \((0,\infty)\). Various applications of such results for Bessel, Kummer and generalized hypergeometric functions are pointed out.

MSC:

26A51 Convexity of real functions in one variable, generalizations
33C20 Generalized hypergeometric series, \({}_pF_q\)
33C15 Confluent hypergeometric functions, Whittaker functions, \({}_1F_1\)
33C05 Classical hypergeometric functions, \({}_2F_1\)
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