[Paper Review] Some hypergeometric summation theorems and reduction formulas via Laplace transform method

M. I. Qureshi, Showkat Ahmad Dar|arXiv (Cornell University)|Aug 17, 2018

Mathematical functions and polynomials3 references3 citations

TL;DR

This paper derives new hypergeometric summation theorems and reduction formulas for generalized hypergeometric functions ${}_{3}F_{2}(/pm 1)$, ${}_{4}F_{3}(1)$, ${}_{6}F_{5}(/pm 1)$, ${}_{7}F_{6}(/pm 1)$, and ${}_{8}F_{7}(/pm 1)$ using the Laplace transform method. It establishes analytical solutions of generalized hyperbolic integrals in terms of these functions, yielding new closed-form results and extending known summation identities with rigorous convergence conditions.

ABSTRACT

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.

Motivation & Objective

• To derive analytical solutions of generalized hyperbolic integrals using the Laplace transform method.
• To obtain new summation theorems for ${}_{3}F_2(-1)$, ${}_{6}F_5(/pm 1)$, ${}_{7}F_6(/pm 1)$, and ${}_{8}F_7(/pm 1)$ not previously recorded in the literature.
• To establish reduction formulas for ${}_{4}F_3(1)$ and ${}_{7}F_6(-1)$ using algebraic properties of Pochhammer symbols.
• To express definite integrals of quotients of hyperbolic functions in terms of generalized hypergeometric functions with explicit convergence criteria.
• To extend known results in hypergeometric function theory by incorporating Laplace transform techniques and product identities of hyperbolic functions.

Proposed method

• Applying the Laplace transform to generalized hyperbolic integrals to express them in terms of hypergeometric functions ${}_{p}F_q(/pm 1)$ with $p, q$ up to 8.
• Using algebraic identities of Pochhammer symbols $(ullet)_n$ to manipulate and reduce hypergeometric series.
• Employing contiguous function relations and integral representations involving the incomplete beta function $B_z(ullet,ullet)$ to derive recurrence-type identities.
• Applying product formulas for hyperbolic functions (e.g., $ anh$, $ anh$, $ anh$) to transform integrands into forms amenable to Laplace transform evaluation.
• Utilizing known summation theorems such as Dixon’s theorem and extending them via parameter substitutions and functional transformations.
• Validating convergence conditions through real parts of parameters, particularly $ ext{Re}(b) > 0$ and $ ext{Re}(b imes ext{sign}(ullet) imes ext{sum}) > 0$, ensuring analyticity.

Experimental results

Research questions

• RQ1 Can Laplace transform techniques be systematically applied to evaluate generalized hyperbolic integrals involving quotients of $ anh$, $ anh$, and $ anh$ functions?
• RQ2 What new summation theorems can be derived for ${}_{3}F_2(-1)$, ${}_{6}F_5(/pm 1)$, ${}_{7}F_6(/pm 1)$, and ${}_{8}F_7(/pm 1)$ using algebraic manipulation of Pochhammer symbols?
• RQ3 How can reduction formulas for ${}_{4}F_3(1)$ and ${}_{7}F_6(-1)$ be constructed using contiguous function relations and integral identities?
• RQ4 What are the precise convergence conditions under which these hypergeometric representations remain valid for integrals involving $rac{ anh(ax) anh(bx)}{ anh(cx)}$ and similar forms?
• RQ5 Can new closed-form evaluations of definite integrals involving products of hyperbolic functions be derived using the Laplace transform and hypergeometric function identities?

Key findings

• The integral $igint_{0}^{ty} rac{ anh(ax) anh(bx)}{ anh(cx)} dx$ is evaluated as a combination of Gamma functions and a ${}_{7}F_6(1)$ hypergeometric function under $ ext{Re}(b) > 0$ and $ ext{Re}(b imes ext{sign}(ullet) imes ext{sum}) > 0$.
• The integral $igint_{0}^{ty} rac{ anh(ax) anh(bx)}{ anh(cx)} dx$ is also expressed in terms of a ${}_{8}F_7(-1)$ function with parameters involving $rac{1}{2} imes ( ext{sign}(ullet) imes ext{radical})$ and convergence ensured by $ ext{Re}(b) > 0$ and $ ext{Re}(b imes ext{sign}(ullet) imes ext{sum}) > 0$.
• A new summation theorem for ${}_{3}F_2(-1)$ is derived: ${}_{3}F_2ig(egin{smallmatrix}a, 1+rac{a}{2}, b \ rac{a}{2}, 1+a-b \ ot ext{end} ight) = rac{ ext{B}(rac{1+a}{2}, 1+a-b)}{ ext{B}(rac{1+a}{2}-b, 1+a)}$ for $ ext{Re}(b) < rac{1}{2}$.
• Reduction formulas are obtained for ${}_{4}F_3(1)$ and ${}_{7}F_6(-1)$, extending known results through parameter transformations and contiguous relations.
• New closed-form evaluations of integrals like $igint_{0}^{ty} rac{ anh(ax) anh(bx)}{ anh(cx)} dx$ are expressed in terms of trigonometric and hyperbolic functions, such as $rac{ an(rac{ heta}{2}) + an(rac{ heta'}{2})}{4b}$, under specified convergence domains.
• The paper establishes that integrals involving $rac{ anh(ax) anh(bx)}{ anh^v(cx)}$ can be evaluated via Gamma functions and ${}_{p}F_q( ext{argument})$ functions with $ ext{Re}(v) < 2$ or $ ext{Re}(v) < 1$, depending on the form.