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Research article Β· 2018 Β· Full text

International Journal of Statistics and Applied Mathematics

MSM fractional integration and differentiation operators of multi-parametric K-Mittag Leffler function and generalized multi-index Bessel function

Mridula PurohitAditi Badguzer

Volume 3(3) Β· Pages 156–161

Research areasFractional Differential Equations SolutionsMathematical Inequalities and ApplicationsNumerical methods in inverse problems

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Abstract

Author-written

In this paper, we aim at finding image of product of Multi-Parameter k-Mittag-Leffler Function and Generalized Multi-Index Bessel Function under the Marichev-Saigo-Maida (MSM) Fractional Integration Operators in terms of generalized k-Wright Function. Some useful special cases of our main results are also obtained.

Summary

Source-checked

Plain-language reading of the full paper, prepared with AI and checked against the text. Each point links to the authors' own words.

  1. Why it matters

    • The study focuses on determining the image of the product of a Multi-Parameter k-Mittag-Leffler function and a Generalized Multi-Index Bessel function when subjected to Marichev-Saigo-Maida (MSM) fractional integration operators. View evidence β†—

      Why it matters

      The study focuses on determining the image of the product of a Multi-Parameter k-Mittag-Leffler function and a Generalized Multi-Index Bessel function when subjected to Marichev-Saigo-Maida (MSM) fractional integration operators.

      In the paper's own words

      β€œIn this paper, we aim at finding image of product of Multi-Parameter k-Mittag-Leffler Function and Generalized Multi-Index Bessel Function under the Marichev-Saigo-Maida (MSM) Fractional Integration Operators in terms of generalized k-Wright Function.”

      Page 1

      Found in: Generalized Fractional Integration and Differentiation of Product of Multi-Parameter K-Mittag-Leffler Function and Generalized Multi-Index Bessel Function

      Read it in the full textOpen PDF at page 1 β†—
    • The authors intend to derive specific special cases from their primary mathematical results to demonstrate their utility. View evidence β†—

      Why it matters

      The authors intend to derive specific special cases from their primary mathematical results to demonstrate their utility.

      In the paper's own words

      β€œSome useful special cases of our main results are also obtained.”

      Page 1

      Found in: Special Cases of Theorems

      Read it in the full textOpen PDF at page 1 β†—

  2. The question

    • The research question is whether the product of the Multi-Parameter k-Mittag-Leffler and Generalized Multi-Index Bessel functions can be expressed in terms of the generalized k-Wright function under MSM operators. View evidence β†—

      The question

      The research question is whether the product of the Multi-Parameter k-Mittag-Leffler and Generalized Multi-Index Bessel functions can be expressed in terms of the generalized k-Wright function under MSM operators.

      In the paper's own words

      β€œIn this paper, we aim at finding image of product of Multi-Parameter k-Mittag-Leffler Function and Generalized Multi-Index Bessel Function under the Marichev-Saigo-Maida (MSM) Fractional Integration Operators in terms of generalized k-Wright Function.”

      Page 1

      Found in: Generalized Fractional Integration and Differentiation of Product of Multi-Parameter K-Mittag-Leffler Function and Generalized Multi-Index Bessel Function

      Read it in the full textOpen PDF at page 1 β†—

  3. What they did

    • The study employs Marichev-Saigo-Maida (MSM) fractional integration and differential operators, which generalize hypergeometric fractional operators using the Appell function $F_3$ in the kernel. View evidence β†—

      What they did

      The study employs Marichev-Saigo-Maida (MSM) fractional integration and differential operators, which generalize hypergeometric fractional operators using the Appell function $F_3$ in the kernel.

      In the paper's own words

      β€œSaigo operators [3] which is an useful generalization of the hyper-geometric fractional operators has been introduced by Marichev [7]... and later extended and studied by Saigo and Maeda [10] in terms of any complex order with Appell function 𝐹3(. ) in the kernel”

      Page 2

      Found in: Generalized K-Wright Function

      Read it in the full textOpen PDF at page 2 β†—
    • The mathematical proofs are conducted by applying function definitions and changing the order of integration and summation. View evidence β†—

      What they did

      The mathematical proofs are conducted by applying function definitions and changing the order of integration and summation.

      In the paper's own words

      β€œProof: Using definitions (1.1), (1.3) & (1.10) and changing the order of integration and summation in LHS of (2.1) it reduces to”

      Page 3

      Found in: Generalized K-Wright Function

      Read it in the full textOpen PDF at page 3 β†—

  4. What they found

    • The authors found that by setting $k=1, p=q=1, a_1 = 𝜏,$ and $b_1 = 1$, the multi-parameter k-Mittag-Leffler function reduces to the generalized Mittag-Leffler function. View evidence β†—

      What they found

      The authors found that by setting $k=1, p=q=1, a_1 = 𝜏,$ and $b_1 = 1$, the multi-parameter k-Mittag-Leffler function reduces to the generalized Mittag-Leffler function.

      In the paper's own words

      β€œAssuming k = 1, p = q = 1, a1 = 𝜏 and b1 = 1 in the main result (2.1) multi-parameter k-Mittag Leffler function reduces to generalized Mittag- Leffler studied by [6]”

      Page 4

      Found in: Special Cases of Theorems

      Read it in the full textOpen PDF at page 4 β†—
    • The results show that setting $C = -1$ and $d = 1$ reduces the Generalized Multi-Index Bessel function to the version obtained by Choi & Agarwal. View evidence β†—

      What they found

      The results show that setting $C = -1$ and $d = 1$ reduces the Generalized Multi-Index Bessel function to the version obtained by Choi & Agarwal.

      In the paper's own words

      β€œIf we put C = -1 & d = 1 in our main result (2.2) then Multi-Index Bessel Function reduced to Multi-Index Bessel Function obtained by Choi & Agarwal [1]”

      Page 5

      Found in: Special Cases of Theorems

      Read it in the full textOpen PDF at page 5 β†—

  5. What it means

    • The authors conclude that they have successfully derived the images of the product of Multi-Parameter k-Mittag-Leffler and Multi-Index Bessel functions under MSM operators in terms of the generalized k-Wright function. View evidence β†—

      What it means

      The authors conclude that they have successfully derived the images of the product of Multi-Parameter k-Mittag-Leffler and Multi-Index Bessel functions under MSM operators in terms of the generalized k-Wright function.

      In the paper's own words

      β€œIn this paper, we studied and obtained images of product of two generalized function namely Multi-Parameter k-Mittag-Leffler Function and Multi-Index Bessel Function under MSM Fractional Integral and Derivatives formulas in terms of generalized k-Wright Function.”

      Page 6

      Found in: Generalized Fractional Integration and Differentiation of Product of Multi-Parameter K-Mittag-Leffler Function and Generalized Multi-Index Bessel Function

      Read it in the full textOpen PDF at page 6 β†—
    • The authors state that their main results are comprehensive enough to encompass several previously published findings as special cases. View evidence β†—

      What it means

      The authors state that their main results are comprehensive enough to encompass several previously published findings as special cases.

      In the paper's own words

      β€œWe obtained many results of earlier work as special cases of our main results.”

      Page 6

      Found in: Special Cases of Theorems

      Read it in the full textOpen PDF at page 6 β†—

Visual summary

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MSM Operator Transformation Flow
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AI-generated conceptual illustration (Abstract mathematical symbols and geometric representations of functions). It explains the idea and does not show study data.

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MSM Operator Transformation Flow

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FULL TEXT Original article Β· reflowed for reading

Introduction Generalized Multi-Index Bessel Function

A new generalization of Bessel function called Multi-Index Bessel Function is introduced by Nisar. K.S. [8] as 𝐽 (πœ‡ 𝑑 ) 𝑠 ,πœ”,𝑑 (𝑣 𝑑 ) 𝑠 ,πœ†,𝐢 (𝑍) = βˆ‘ (1.1)

For 𝑣 𝑑 , πœ‡ 𝑑 , πœ†, 𝑑, 𝑐 πœ– 𝐢 (t = 1, 2 …s) be such that

Re (𝜈 𝑑 ) > max{0, 𝑅𝑒(πœ”) -1}; πœ” > 0, R𝑒(πœ‡ 𝑑 ) > 0 and Re (πœ†) > 0, & s πœ– N Here (πœ†) πœ”π‘™ denotes the Pochhammer symbol defined (for πœ†, πœ”π‘™ πœ– 𝐢 ) in terms of Gamma function Ξ“ (see [12] , section 1 Multi-Parameter K-Mittag-Leffler Function Gehlot K.S. introduced Multi-Parameter k-Mittag-Leffler function [4] as (1.3)

Where Ξ“ π‘˜ (π‘₯) is the generalized k -Gamma function [2] given by

And (π‘₯) 𝑛,π‘˜ is the k-pochhammer symbol defined as

For Re(x) > 0, Ξ“ k (x) is defined as the integral

Generalized K-Wright Function

For Z, π‘Ž 𝑖 , 𝑏 𝑗 πœ– 𝑐 and 𝛼 𝑗 , 𝛽 𝑗 πœ– R (𝛼 𝑗 , 𝛽 𝑗 β‰  0 ; 𝑖 = 1, 2, . . 𝑝; j = 1,2, …q) the generalized k-Wright Function [3] is represented as

Gehlot. K.S. and Prajapati [5] defined the generalized k-Wright Function as Ξ¨ π‘ž π‘˜ 𝑝 (𝑧) forπ‘Ž 𝑖 ,𝑏 𝑗 , z πœ– 𝐢, k πœ– 𝑅 + , 𝛼 𝑖 , 𝛽 𝑗 ∈ 𝑅(𝛼 𝑖 , 𝛽 𝑗 β‰  0; 𝑖 = 1,2, … . 𝑝 ; 𝑗 = 1, 2, … . π‘ž) π‘Žπ‘›π‘‘ (π‘Ž 𝑖 + 𝛼 𝑖 𝑛), (𝑏 𝑗 + 𝛽 𝑗 𝑛) πœ– 𝐢 \π‘˜π‘ Μ… , as

Marichev-Saigo-Maida (Msm) Fractional Integration and Differential Operators Saigo operators [3] which is an useful generalization of the hyper-geometric fractional operators has been introduced by Marichev [7] (see details in samko et al. [11] , P-194, (10.47) and whole section 10.3) and later extended and studied by Saigo and Maeda [10] in terms of any complex order with Appell function 𝐹 3 (. ) in the kernel as follows:

) 𝑓(𝑑)𝑑𝑑, Re(𝛾) > 0 (1.11) These operators (1.10)-(1.13) reduce to the Saigo fractional calculus operators [9] as

The left-hand sided and right-hand sided generalized integration of the type (1.10) and (1.11) for a power function formula | see [11] , P-394, eq (4.18) and (4.19) are given by

where Re(𝛾) > 0,Re(𝜎) < 1+ min { Re(-𝛽), Re(𝛼 + 𝛼 β€² -𝛾), 𝑅𝑒(𝛼 + 𝛽 β€² -𝛾)}.

Generalized Fractional Integration and Differentiation of Product of Multi-Parameter K-Mittag-Leffler Function and Generalized Multi-Index Bessel Function

Here, in this section we find images of product of Multi-Parameter k-Mittag Leffler Function and Generalized Multi-Index Bessel Function under the MSM operators in terms of generalized k-Wright Function. These formulas are given by the following theorem. .

Using (1.7) and definition of generalized k-Wright Function from (1.9) we get the desired result.

Theorem 2

Let 𝛼, 𝛼 β€² , 𝛽, 𝛽 β€² , 𝛾, 𝜎, 𝑣 𝑑 , 𝑒 𝑑 , πœ†, 𝑑, 𝑐, π‘Ž 𝑗 , 𝑏 𝑗 , πœ– 𝐢, π‘˜ > 0, 𝛼 𝑗 , 𝛽 𝑗 , ∈ 𝑅 and π‘₯ > 0 be such that βˆ‘ 𝑅𝑒(𝑣 𝑑 ) > max {0, 𝑅𝑒(π‘˜) -1}; π‘˜ > 0, 𝑅𝑒(πœ‡ 𝑑 ) > 0, Re(πœ†) > 0

Proof: Similar to as of Theorem 1 & 2.

Special Cases of Theorems

(i) Assuming k = 1, p = q = 1, a 1 = 𝜏 and b 1 = 1 in the main result (2.1) multi-parameter k-Mittag Leffler function reduces to generalized Mittag-Leffler studied by [6], thus we have Where (2.7) is a known result obtained by S.D. Purohit, D.L. Suthar and S.L. Kalla in [14] Special Cases of Theorem 2 (i) Taking k = 1, p = q = 1, a 1 = 𝜏 and b 1 = 1 in the main result (2.2) Multi-Parameter k-Mittag-Leffler Function reduced to generalized Mittag-Leffler studied by [6] , thus we have (2.8)

(ii) If we put k = 1, p = q = m= 1, a 1 = b 1 = 1 in our main result (2.2) Multi-Parameter k-Mittag-Leffler Function reduced to Mittag-Leffler studied by [13] , thus we have

(1 -𝜎 -𝜌 β€² 𝑙 -𝛾 + 𝛼 + 𝛽 β€² , 𝜌);

(1 -𝜎 -𝜌 β€² 𝑙 -π‘˜π›½ + π‘˜π›Ό, 𝜌);

(iii) If we put C = -1 & d = 1 in our main result (2.2) then Multi-Index Bessel Function reduced to Multi-Index Bessel Function obtained by Choi & Agarwal [1] , so we have

References

12 in the original paper
  1. Choi J, Agarwal P. A note on fractional integral operator associated multiindex Mittag-Leffler functions, Filomat. 2016; 30(7):1931-1939.
  2. Diaz R, Pariguan E. On hypergeometric functions and Pochhammer k-symbol. Divulgaciones Mathematicas. 2007; 15(2):179-192.
  3. Wright EM. The asymptotic expansion of generalized hypergeometric Function. J London Math. Soc. 1935; 10:286-293. Doi: 10.1112/jlms/s1-10.40.286.
  4. Gehlot KS. Multiparameter K-Mittag-Leffler Function,International Mathematical Forum. 2013; 8(34):1691-1702.
  5. Gehlot Prajapati KS. JC. On generalization of K-Wright function andit's properties, International Journal of Pure and Applied Mathematics. 2014; 93(5):715-728.
  6. Marichev OI. Volterra equation of Mellin Convolution type with a Horn function in the kernel (In Russian), Izv. ANBSSR Ser. Fiz.-Mat. Nauk. 1974; 1:128-129.
  7. Nisar KS. Fractional calculus and certain integrals of generalized multiindex bessel function, Mathematics Subject Classification, 2017, 33C20; 33B15.
  8. Saigo M. A remark on integral operators involving the Gauss hypergeometric functions, Math. Rep. Kyushu Univ. 1978; 11:135-143.
  9. Saigo M, Maeda N. More generalization of fractional calculus, Transform Methods and Special Functions, Varna, Bulgaria, 1996, 386-400.
  10. Samko SG, Kilbas AA, Marichev OI. Fractional Integrals and Derivatives. Theory and Applications, Gordan and Breach Sci. Publ. New York, 1993.
  11. Wiman A. Uber den fundamental Satz in der Theories der Funktionen EΞ± (z), Acta Math. 1905; 29:191-201.
  12. Purohit SD, Suthar DL, Kalla SL. Marichev-Saigo-Maeda Fractional Integration Operators of the Bessel Functions LE Matematiche. 2012; LXVII(I):21-32

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