arXiv subjects

Showkat Ahmad Dar

Publications and source records attributed to Showkat Ahmad Dar.

3 recordsLinked to original sources

arXiv2018

Some hypergeometric summation theorems and reduction formulas via Laplace transform method

In this paper, we obtain analytical solutions of Laplace transform based some generalized class of the hyperbolic integrals in terms of hypergeometric functions ${}_3F_2 (\pm1)$, ${}_4F_3 (\pm1)$, ${}_5F_4(\pm1)$, ${}_6F_5(\pm1)$, ${}_7F_6(\pm1)$ and ${}_8F_7(\pm1)$ with suitable convergence conditions, by using some algebraic properties of Pochhammer symbols. In addition, reduction formulas for ${}_4F_3(1)$, ${}_7F_6(-1)$ and some new summation theorems (not recorded earlier in the literature of hypergeometric functions) for ${}_3F_2(-1)$, ${}_6F_5(\pm1)$, ${}_7F_6(\pm1)$ and ${}_8F_7(\pm1)$ are obtained.

math.CA↗

arXiv2018

Laplace transforms based some novel integrals via hypergeometric technique

In this paper, we obtain the analytical solutions of Laplace transforms based some novel integrals with suitable convergence conditions, by using hypergeometric approach (some algebraic properties of Pochhammer symbol and classical summation theorems of hypergeometric series ${}_{2}F_{1}(1)$, ${}_{2}F_{1}(-1)$ , ${}_{4}F_{3}(-1)$) . Also, we obtain the Laplace transforms of arbitrary powers of some finite series containing hyperbolic sine and cosine functions having different arguments, in terms of hypergeometric and Beta functions. Moreover, Laplace transforms of even and odd positive integral powers of sine and cosine functions with different arguments, and their combinations of the product (taking two, three, four functions at a time), are obtained. In addition, some special cases are yield from the main results.

math.CA↗

arXiv2018

Generalizations of Ramanujan integral associated with infinite Fourier cosine transforms in terms of hypergeometric functions and its applications

In this paper, we obtain analytical solution of an unsolved integral $\textbf{R}_{C}(m,n)$ of Srinivasa Ramanujan [$\textit{Mess. Math}$., XLIV, 75-86, 1915], using hypergeometric approach, Mellin transforms, Infinite Fourier cosine transforms, Infinite series decomposition identity and some algebraic properties of Pochhammer's symbol. Also we have given some generalizations of the Ramanujan's integral $\textbf{R}_{C}(m,n)$ in the form of integrals $\textbf{I}^{*}_{C}(\upsilon,b,c,λ,y), \textbf{J}_C (\upsilon,b,c,λ,y), \textbf{K}_{C} (\upsilon,b,c, λ,y), \textbf{I}_{C}(\upsilon,b,λ,y)$ and solved it in terms of ordinary hypergeometric functions ${}_2 F_3$, with suitable convergence conditions. Moreover as applications of Ramanujan's integral $\textbf{R}_{C}(m,n)$, the new nine infinite summation formulas associated with hypergeometric functions ${}_{0}F_{1}$, ${}_{1}F_{2}$ and ${}_{2}F_{3}$ are obtained.

math.CA↗