GENERALIZED FRACTIONAL DERIVATIVE OPERATORS OF THE PRODUCT OF MULTI-INDEX BESSEL FUNCTION AND MULTI-

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Volume: 12 Issue: 03 | Mar 2025

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GENERALIZED FRACTIONAL DERIVATIVE OPERATORS OF THE PRODUCT OF MULTI-INDEX BESSEL FUNCTION AND MULTI-INDEX MITTAG LEFFLER FUNCTION WITH APPLICATIONS Krishna Gopal Bhadana1 and Sunil Kumar2 1, 2 Department of Mathematics, S. P. C. Government College, Ajmer

Maharshi Dayanand Saraswati University, Ajmer, Rajasthan-305009, India ---------------------------------------------------------------------------***--------------------------------------------------------------------------Abstract In the present paper we have introduced multi-index Bessel function and multi-index Mittag Leffler function and established the generalized fractional derivative operators of the product of generalized multi-index Bessel function and multi-index Mittag Leffler function. Further, the Riemann-Liouville, fractional derivative operators of given functions are obtained. 2020 Mathematics Subject Classification: 26A33, 33E12, 33C10.

Keywords and Phrases: generalized fractional derivative operators, multi-index Bessel function, multi-index Mittag Leffler function.

Definitions 1 Generalized Multi-Index Bessel Function For 𝐴𝑗 , 𝐵𝑗 , 𝜆, ∈ ℂ, µ > 0, ℛ𝑒(𝜆) > 0 the generalized multi-index Bessel function is defined by Choi and Agarwal ,1- in the following summation form: ( 𝐴𝑗 )

𝑚,𝜆

𝙹( 𝐵 )

𝑗 𝑚,µ

∞

(𝑥) = ∑ 𝑙=0

(−𝑥)𝑙 ; 𝑙! ∏𝑚 𝑗=1 𝛤(𝐴𝑗 𝑙 + 𝐵𝑗 + 1) (𝜆)𝜇𝑙

(𝑚 ∈ ℕ)

(1.1)

where ℛ𝑒(𝐵𝑗 ) > −1 and ∑𝑚 𝑗=1 ℛ𝑒(𝐴𝑗 ) > 𝑚𝑎𝑥 {0; ℛ𝑒(𝜇) – 1}

2 Generalized Multi-Index Mittag Leffler Function For 𝐴𝑗 , 𝐵𝑗 , 𝜆, 𝜌 ∈ ℂ, the generalized multi-index Mittag Leffler function is defined by Saxena and Nishimoto ,13- in the following summation form ∞ (𝜆)𝜌𝑘 𝑥𝑘 𝜆,𝜌 (2.1) 𝐸(𝐴 ,𝐵 ) (𝑥) = ∑ 𝑚 ; (𝑚 ∈ ℕ) 𝑗 𝑗 𝑚 ∏𝑗=1 𝛤(𝐴𝑗 𝑘 + 𝐵𝑗 ) 𝑘! where ℛ𝑒(𝐵𝑗 ) > 0 and ∑𝑚 𝑗=1 ℛ𝑒(𝐴𝑗 ) > 𝑚𝑎𝑥 {0; ℛ𝑒(𝜌) – 1; 0}.

𝑘=0

For 𝑚 = 1 the generalized multi-index Mittag Leffler function (2.1) reduce into the generalized Mittag-Leffler function given by Shukla and Prajapati ,16- and defined as ∞ 𝜆,𝜌

𝛦𝐴,𝐵 (𝑥) = ∑ 𝑘=0

𝑥𝑘 , 𝛤(𝐴𝑘 + 𝐵) 𝑘! (𝜆)𝜌𝑘

(2.2)

where 𝐴, 𝐵, 𝜆 ∈ ℂ; ℛ𝑒(𝐴) > 0, ℛ𝑒(𝐵) > 0, ℛ𝑒(𝜆) > 0 and 𝜌 ∈ (0, 1) ∪ ℕ For 𝑚 = 1 and 𝜌 = 1, the generalized multi-index Mittag Leffler function (2.1) reduce into the generalized Mittag-Leffler function given by Prabhakar ,10- defined as

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