Abstract
Motivated by recent generalizations of classical theorems for the series 2F1 [Integral Transform. Spec. Funct. 229(11), (2011), 823-840] and interesting Laplace transforms of Kummer's confluent hypergeometric functions obtained by Kim et al. [Math. Comput. Modelling 55 (2012), 1068-1071], first we express generalized summations theorems in explicit forms and then by employing these, we derive various new and useful Laplace transforms of convolution type integrals by using product theorem of the Laplace transforms for a pair of Kummer's confluent hypergeometric function.
Highlights
AND PRELIMINARIESWe begin with the definition of the generalized hypergeometric function with p numerator and q denominator parameters is defined by [7] (1)pFq a1, . . . , ap b1, . . . , bq z = pFq (a) (b) z :=(a1)m · · ·m zm, m≥0 (b1)m · · ·m m!where the Pochhammer symbol is defined by (c)0 = 1, (c)n = c(c + 1) · · · (c + n − 1), and bj ∈ C\Z−0, j = 1, s := 1, 2, . . . , s
The importance of the generalized hypergeometric function lies in the fact that almost all elementary functions such as exponential, binomial, trigonometric, hyperbolic, logarithmic etc. are special case of this function
First we provide the explicit expressions for all pairs of coefficients {Ai(a, b), Bi(a, b)}, {Ci(a, b), Di(a, b)} and {Ei(a, b), Fi(a, b)} in (5), (6) and (7), respectively, thereafter, we demonstrate how one can obtain Laplace transforms of convolution type integral by using product theorem for Laplace transforms for a pair of Kummer’s confluent hypergeometric function by employing generalized summation formulas (5), (6) and (7)
Summary
We begin with the definition of the generalized hypergeometric function with p numerator and q denominator parameters is defined by [7]. The well known classical summation theorems such as those of Gauss second summation theorem, Bailey summation theorem and Kummer’s summation theorem for the series 2F1 which are given below, play an important role in the theory of hypergeometric functions. First we provide the explicit expressions for all pairs of coefficients {Ai(a, b), Bi(a, b)}, {Ci(a, b), Di(a, b)} and {Ei(a, b), Fi(a, b)} in (5), (6) and (7), respectively, thereafter, we demonstrate how one can obtain Laplace transforms of convolution type integral by using product theorem for Laplace transforms for a pair of Kummer’s confluent hypergeometric function by employing generalized summation formulas (5), (6) and (7). By using product theorem for Laplace transform for a pair of Kummer’s confluent hypergeometric functions, presented shortly, in Section 4 we employ the generalized summation formulas (5), (6) and (7) in order to derive new Laplace transforms of convolution type integrals involving 1F1(a; b; x).
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