[Paper Review] Some unified integrals associated with Bessel-Struve kernel function

Kottakkaran Sooppy Nisar, Praveen Agarwal+1 more|arXiv (Cornell University)|Jan 31, 2016

Mathematical functions and polynomials13 references6 citations

TL;DR

This paper derives unified integral formulas involving the Bessel-Struve kernel function $ S_{eta}( u z) $, expressing them in terms of generalized Wright functions. By leveraging series expansions of the kernel and applying a known integral identity, the authors establish new closed-form results that generalize previous integrals of Bessel and Struve functions, with special cases recovering known identities for $ I_0, L_0, I_1, L_1 $.

ABSTRACT

In this paper, we discuss the generalized integral formula involving Bessel-Struve kernel function $S_{α}\left( λz ight) $, which expressed in terms of generalized Wright functions. Many interesting special cases also obtained in this study.

Motivation & Objective

• To derive new unified integral formulas involving the Bessel-Struve kernel function $ S_{eta}( u z) $.
• To express these integrals in terms of generalized Wright hypergeometric functions $ {}_p\Psi_q $.
• To recover known integral formulas for Bessel and Struve functions as special cases by choosing specific values of the parameter $ \beta $.
• To extend previous results on unified integrals of Bessel functions to the broader class of Bessel-Struve functions.

Proposed method

• Utilizes the power series representation of the Bessel-Struve kernel function $ S_{\alpha}(\lambda z) = \sum_{n=0}^{\infty} \frac{(\lambda z)^n \Gamma(\alpha+1) \Gamma((n+1)/2)}{\sqrt{\pi} n! \Gamma(n/2 + \alpha + 1)} $.
• Applies the integral identity $ \int_0^\infty x^{\mu-1} (x+a+\sqrt{x^2+2ax})^{-\lambda} dx = 2^{1-\mu} a^{\mu-\lambda} \frac{\Gamma(2\mu) \Gamma(\lambda - \mu)}{\Gamma(1+\lambda+\mu)} $ for $ 0 < \text{Re}(\mu) < \text{Re}(\lambda) $.
• Interchanges summation and integration under uniform convergence conditions to derive results in terms of generalized Wright functions.
• Expresses the final results using the generalized Wright function $ {}_3\Psi_2 $ and $ {}_2\Psi_2 $, with specific parameter sets derived from the kernel's series.
• Uses known relations between Bessel-Struve kernel and Bessel/Struve functions, such as $ S_0(z) = I_0(z) + L_0(z) $ and $ S_1(z) = (2I_1(z) + L_1(z))/z $, to recover special cases.

Experimental results

Research questions

• RQ1 Can unified integral formulas for the Bessel-Struve kernel function be derived using generalized Wright functions?
• RQ2 How do the generalized Wright functions emerge in the evaluation of integrals involving $ S_{\alpha}(\nu z) $?
• RQ3 What are the special cases of the derived integral formulas when $ \alpha = 0 $ or $ \alpha = 1 $, corresponding to combinations of $ I_0, L_0, I_1, L_1 $?
• RQ4 In what way do the results generalize known integral formulas for Bessel and Struve functions?

Key findings

• The integral $ \int_0^\infty x^{\mu-1} (x+a+\sqrt{x^2+2ax})^{-\lambda} S_{\alpha}(\gamma y / (x+a+\sqrt{x^2+2ax})) dx $ evaluates to $ 2^{1-\mu} a^{\mu-\lambda} \frac{\Gamma(\alpha+1) \Gamma(2\mu)}{\sqrt{\pi}} \times {}_3\Psi_2\left[ \cdots \right] $.
• For $ \alpha = 0 $, the result reduces to an integral involving $ I_0(y/\cdot) + L_0(y/\cdot) $, expressed as $ {}_3\Psi_3 $ with specific parameters.
• For $ \alpha = 1 $, the result corresponds to $ 2I_1(y/\cdot) + L_1(y/\cdot) $, yielding a $ {}_2\Psi_2 $ expression.
• The derived formulas generalize known integrals of Bessel and Struve functions by unifying them under the Bessel-Struve kernel framework.
• The conditions $ \text{Re}(\mu) > 0 $, $ \text{Re}(\lambda) > \text{Re}(\mu) $, and $ x > 0 $ ensure convergence and validity of the results.