Abstract
Motivated mainly by a variety of applications of Euler's Beta, hypergeometric, and confluent hypergeometric functions together with their extensions in a wide range of research fields such asengineering, chemical, and physical problems. In this paper, we introduce modified forms of some extended special functions such as Gamma function, Beta function, hypergeometric function and confluent hypergeometric function by making use of the idea given in reference \cite{9}. Also, certain investigations including summation formulas, integral representations and Mellin transform of these modified functions are derived. Further, many known results are obtained asspecial cases of our main results.
Highlights
The classical Gamma and Beta functions are defined respectively as follows: ∞Γ (x) = tx−1e−tdt, (Re (x) > 0) . (1.1) andB (x, y) = tx−1 (1 − t)y−1 dt, (Re (x) > 0, Re (y) > 0 ), (1.2)for each x, y ∈ (0, +∞)
It is clear that when p = 0, the equations (1.6) and (1.7) reduce to the well known classical hypergeometric and confluent hypergeometric functions respectively
Motivated by the idea given in [1], in this paper, we introduce a new modified forms of the extended Gamma, Beta, hypergeometric and confluent hypergeometric functions defined in equations (1.4)-(1.7) respectively
Summary
The classical Gamma and Beta functions are defined respectively as follows (see [2]):. The extended hypergeometric and confluent hypergeometric functions are introduced in [4] as follows: where Re (p). It is clear that when p = 0 , the equations (1.6) and (1.7) reduce to the well known classical hypergeometric and confluent hypergeometric functions respectively. The modified Laplace transform is introduced by Saif et al [1] as follows: La{f(t)} = a−stf(t)dt, Re(s) > 0, a ∈ (0, ∞)\{1}, which for a = e reduces to the known Laplace transform given as: L{f(t)} = e−stf(t)dt, Re(s) > 0,. Motivated by the idea given in [1], in this paper, we introduce a new modified forms of the extended Gamma, Beta, hypergeometric and confluent hypergeometric functions defined in equations (1.4)-(1.7) respectively
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