EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6279 ISSN 1307-5543 – ejpam.com Published by New York Business Global Generalized Extended Confluent, Whittaker k-Functions and Their Properties Syed Ali Haider Shah1, Mujahid Hussain Shah1, Miguel Vivas-Cortez2,∗, Shahid Mubeen3, Gauhar Rahman4 1 Department of Mathematics, University of Sargodha, Sargodha 40100, Pakistan 2 Pontificia Universidad Católica del Ecuador, Faculty of Exact, Natural and Environmental Sciences, FRACTAL Laboratory (Fractional Research in Analysis, Convexity and Their Applications Laboratory), Ecuador 3 Department of Mathematics, Baba Guru Nanak University, Nankana Sahib 3900, Pakistan 4 Department of Mathematics & Statistics, Hazara University, Mansehra 21300, Pakistan Abstract. The main objective of this research paper is to explore further generalization of conflu- ent hypergeometric and Whittaker functions by introducing a new parameter k > 0, in generalized extended confluent hypergeometric and Whittaker functions defined by Khan et al. [1]. We also investigate the Mellin transformations, inverse Mellin transformations, Hankel transformations, Laplace transformations, and derivative of the newly defined generalized extended confluent hy- pergeometric and Whittaker k-functions. We also obtain Riemann-Liouville fractional integral and Riemann-Liouville k-fractional integral of these new generalized extended Whittaker k-function. 2020 Mathematics Subject Classifications: 33C60, 33C20, 33C05, 33C15 Key Words and Phrases: Gamma k-function 1. Introduction and Preliminaries Special functions play a compelling role in different fields such as statistics, physics, mathematics and engineering etc. The hypergeometric, beta, gamma functions and Leg- endre polynomial play a remarkable role to solve complex mathematical problems. In the solutions of partial differential equation to control the physical phenomena such as wave propagation and heat flow, the special functions are contiguously arrive. The hy- pergeometric functions are used in the calculation of Feymann integrals. The confluent hypergeometric has many applications in probability, statistics, approximation theory, so- lution of differential equation and physics. The confluent hypergeometric function helps ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6279 Email addresses: ali.bukhari78699@gmail.com (S. A. H. Shah), shahmujahi@gmail.com (M. H. Shah), mjvivas@puce.edu.ec (M. Vivas-Cortez), smjhanda@gmail.com (S. Mubeen), drgauhar.rahman@hu.edu.pk, gauhar55uom@gmail.com (G. Rahman) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) S. A. H. Shah et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6279 2 of 23 in the solution of Schrödinger equation in quantum mechanics and the heat conduction equation in the thermodynamics. It also helps in the study of random process, stochastic model statistical distributions, in the calculation of Laplace transform and Fourier trans- form. Nasa uses generalized Whittaker functions to model spacecraft re-entry plasma sheaths. Due to generalization of such hypergeometric and Whittaker functions we can solve closed-form solutions to differential equations with non-polynomial coefficients. In quantum field theory and fractional calculus models higher-order differential equations can be solved by using such generalizations. The generalization and extension of special functions such as hypergeometric, gamma, beta, Bessel functions etc. play a central role in mathematical modeling, economics, physics and mathematical analysis. In mathematical analysis such as integral transforms, and special function theory generalizes the Mellin and Barnes integral representations for functions with branch cuts. Various researches introduced the generalizations, extensions, integral representations and properties of various special functions for parameter k > 0, (see [2–7] and [8–14]). In [15], Diaz and Pariguan investigated gamma, beta, hypergeomet- ric k-functions and Pochhammer’s k-symbol. Nasar et al.[16], introduced some inequalities involving extended gamma and confluent hypergeometric k-functions. In [17], Mubeen et al. investigated the extensions of beta, gamma, and beta distribution. In [18], Qayyumm et al. introduced extended conformable k- beta and hypergeometric functions by using Mittag-Leffer k-function. In [19], Kokologiannaki proved some properties and inequalities of gamma, beta and zeta k- functions. In [20], Mubeen et al. introduced integral rep- resentations of k-hypergeometric functions. Rahman et al. [21] investigated inequalities involving extended gamma and beta k-functions. Many other researchers introduced the generalizations, extensions, integral represen- tations and properties of various special functions without k > 0 parameter, (see [1, 4, 10– 12, 22]). In 2004, Chaudhary et al. [23] extended the Gauss, confluent hypergeometric functions by using the extended beta functions. Further, Parmar [24] introduced a new generalization of extended Gauss, confluent hypergeometric functions by using generalized extended beta functions. In [25], Ozregion et al. gave the extension of gamma, beta, and hypergeometric functions. Sarivastava et al. [26] introduced a new generalized extended Gauss hypergeometric functions. Issenova et al. [27] gave some generalizations of Whit- taker, Horn, Bessel, Legendre functions, discussed some related properties and examples as applications. Paula et al. [28] investigated the generalization of some special functions and discussed related application in probability distributions in the field of statistics. The Whittaker function which was introduced by Whittaker in (1903) is a unique solution of Whittaker equation. It is a modified form of confluent hypergeometric function. It has many applications in physics, engineering and mathematics. Whittaker function helps in the signal processing, and to solve the differential equations. The Whittaker function introduced by whittaker in [29]. After that various researchers introduced the generaliza- tions and extensions of Whittaker function in terms of k > 0 parameter and without k parameter. In [30], Nagar et al. introduced extended Whittaker function and its prop- S. A. H. Shah et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6279 3 of 23 erties. Khan et al. [31] generalized extended whittaker function. In [32], Khan et al. investigated the analysis of extended Whittaker function. In [33], Khan et al. also in- troduced multi-index Whittaker function. In [34], Panwar and Rai introduced Whittaker k-function and investigated the fractional integral of Whittaker k-function. In [1], Khan et al. investigated generalized extended Whittaker function introducing an extra parameter. The main objective of this research paper is to provide a further generalization of confluent hypergeometric and Whittaker functions by introducing k > 0 parameter in generalized extended confluent hypergeometric and Whittaker functions defined by Khan et al. [1]. We also investigate some properties such as integral representations, Mellin transform, inverse Mellin, Hankel, laplace transformations and derivative of these new generalized extended confluent hypergeometric and Whittaker k-functions. We also ob- tain Riemann-Liouville fractional integral and k-Riemann-Liouville fractional integral of these new generalized extended Whittaker k-function. The laplace transformations helps us in solving complex mathematical problems, design and analyze the control systems, analyzing and optimizing communications signals in telecommunications and used in fi- nancial modeling. Mellin and Hankel transformations are important mathematical tools in the field of integral transforms. For the simplifying complex problems, dealing with spe- cial functions, in number theory and probability theory Mellin and Hankel transformations play a key role. In [15], Diaz and Pariguan investigated gamma, beta, hypergeometric k-functions and Pochhammer’s k-symbol as follows: Let w ∈ C (C is a set of complex numbers), then Γk(w) = ∞∫ 0 vw−1e− vk k dv. (1) If ℜ(s1) > 0, ℜ(s2) > 0, k > 0, then βk(s1, s2) = Γk(s1)Γk(s2) Γk(s1 + s2) (2) = 1 k 1∫ 0 s s1 k −1(1− s) s2 k −1ds. (3) If τ, u ∈ C; k > 0, then (λ)u,k = Γk(τ + uk) Γk(τ) (τ ∈ C\{0}) = { 1 (u = 0), τ(τ + k) · · · (τ + (l − 1)k) (u = l ∈ N). (4) The Gauss hypergeometric k-function is defined as 2 F1,k(λ1, λ2;λ3; z) = ∞∑ n=0 (λ1)n,k(λ2)n,k (λ3)n,k zn n! , ( λ3 ∈ C\𭟋− 0 ; |z| < 1; k ∈ R+ ) . (5) S. A. H. Shah et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6279 4 of 23 In [16], confluent hypergeometric k-function is defined as 1ψ1,k(σ1, σ2; t) = ∞∑ m=0 (σ1)m,k (σ2)m,k tm m! , (6) where |t| < 1 k , ℜ(σ1) > ℜ(σ2) > 0, k > 0. In [34], Savita Panwar and Prakriti Rai introduced a new form of confluent hypergeo- metric k-function as 1F1,k(s1, s2; v) = ∞∑ m=0 βk(s1 +mk, s2 − s1) βk(s1, s2 − s1) vm m! . (7) In [2], Mubeen introduced the following k-analague of Kummer’s first formula 1F1,k(s1, s2; v) = exp(v)1F1,k(s2 − s1, s2;−v). (8) In [17], Mubeen et al. defined extended gamma k-function as follows: Γ (p,q) w,k (v) = ∞∫ 0 tv−1 1F1,k(p; q;− tk k − wk ktk )dt, (9) where ℜ(w) > 0, ℜ(q) > 0, ℜ(v) > 0 , ℜ(p) > 0, k > 0. In the same paper [17], extended beta k-function is defined as β (δ1,δ2) ξ,k (p, q) = 1 k 1∫ 0 s p k −1(1− s) q k −1 1F1,k(δ1; δ2; −ξk ks(1− s) )ds, (10) where ℜ(δ1) > 0, ℜ(δ2) > 0, ℜ(ξ) ≥ 0, ℜ(p) ≥ 0, ℜ(q) ≥ 0, k > 0. Let k > 0, α ∈ (0, 1), then extended (α, k)-beta function defined in [18] as βα,Qk,p1,p2 (p, q) = 1 αk 1∫ 0 s p αk −1(1− s) q αk −1E(k,p1,p2) ( −Qk ks(1− s) ) dαs, (11) where ℜ(p),ℜ(q) > 0, s ∈ C, Q ≥ 0. In the same paper [18], (α, k)- hypergeometric and confluent hypergeometric functions respectively, defined as Fα,Q k,p1,p2 (v1, v2, v3;x α) = ∞∑ m=0 (v1)m,kβ α,Q k,p1,p2 (v2 +mkα, v3 − v2) βαk (v2, v3 − v2) xαm m! , (12) S. A. H. Shah et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6279 5 of 23 where ℜ(v1,ℜ(v2),ℜ(v3 > 0), α ∈ (0, 1), k > 0, |xα| < 1, Q ≥ 0 and ϕα,Qk,p1,p2 (v2, v3;x α) = ∞∑ m=0 βα,Qk,p1,p2 (v2 +mkα, v3 − v2) βαk (v2, v3 − v2) xαm m! , (13) where ℜ(v2),ℜ(v3) > 0, α ∈ (0, 1), k > 0, |vα| < 1, Q ≥ 0. By using equation (11) in (12) and (13), we obtain following integral representations Fα,Q k,p1,p2 (v1, v2, v3;x α) = 1 αkβαk (v2, v3 − v2) 1∫ 0 s v2 αk −1(1− s) v3−v2 αk −1(1− kxαs) −v1 k ×E(k,p1,p2) ( −Qk ks(1− s) ) dαs, (14) where ℜ(v1) > 0,ℜ(v2),ℜ(v3) > 0, α ∈ (0, 1), k > 0, |xα| < 1, Q ≥ 0. ϕα,Qk,p1,p2 (v2, v3;x α) = 1 αkβαk (v2, v3 − v2) 1∫ 0 s v2 αk −1(1− s) v3−v2 αk −1ex αs ×E(k,p1,p2) ( −Qk ks(1− s) ) dαs, (15) where ℜ(v2),ℜ(v3) > 0, α ∈ (0, 1), k > 0, |xα| < 1, Q ≥ 0. Khan et al. [1], introduced the generalized extended confluent and beta functions as ψ (δ1,δ2,l1:l2) ξ (σ2, σ3; t) = ∞∑ m=0 β (δ1,δ2,l1,l2) ξ (σ2 +m,σ3 − σ2) β(σ2, σ3 − σ2) tm m! , (16) where |t| < 1,min{ℜ(δ1),ℜ(δ2),ℜ(l1),ℜ(l2)} > 0,ℜ(σ3) > ℜ(σ2) > 0,ℜ(ξ) ≥ 0 l1, l2 ≥ 1, and β (δ1,δ2,l1,l2) ξ (u, v) = 1∫ 0 zu−1(1− z)v−1 1F1(δ1; δ2; −ξ zl1(1− z)l2 )dz. (17) By using equation (17) into (16), we obtain the following integral representation of gener- alized extended confluent hypergeometric function ψ (δ1,δ2,l1:l2) ξ (σ1, σ2, σ3; t) = 1 β(σ2, σ3 − σ2) ∫ 1 0 tσ2−1(1− t)σ3−σ2−1ezt ×1F1(δ1; δ2; −ξ tl1(1− t)l2 )dt. (18) S. A. H. Shah et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6279 6 of 23 Srivastava et al. [26], introduced the generalized extended Gauss hypergeometric as follow: F (δ1,δ2,l1:l2) ξ (σ1, σ2, σ3; t) = ∞∑ m=0 (σ1)m β (δ1,δ2,l1,l2) ξ (σ2 +m,σ3 − σ2) β(σ2, σ3 − σ2) tm m! , (19) where |t| < 1,min{ℜ(δ1),ℜ(δ2),ℜ(l1),ℜ(l2)} > 0,ℜ(σ3) > ℜ(σ2) > 0,ℜ(σ1) > 0, ω ≥ 0, l1, l2 ≥ 1. By using equation (17) into (19), we obtain the following integral representation of generalized extended confluent hypergeometric function F (δ1,δ2,l1:l2) ξ (σ1, σ2, σ3; t) = 1 β(σ2, σ3 − σ2) ∫ 1 0 tσ2−1(1− t)σ3−σ2−1 ×(1− zt)−σ1 1F1(δ1; δ2; −ξ tl1(1− t)l2 )dt. (20) Khan et al. [1] introduced the extension of Kummer’s first formula as follows: ψ (δ1,δ2,l1:l2) ξ (σ2, σ3; t) = exp(t)ψ (δ1,δ2,l1;l2) ξ (σ3 − σ2, σ3;−t). (21) Remark 1. If we take l1 = l2 into (16), (17), (18), (19), (20), we get the extended Gauss, confluent and beta function and their integral representations respectively which introduced by Parmar [24]. Further if we take σ1 = σ2 and l1 = l2 = 1 into (16), (17), (18), (19), (20), we get the extended Gauss, confluent, beta functions and their integral representations respectively which introduced by Chaudhry et al. [23]. Further if we take l1 = l2 = 1 into (16), (17), (18), (19), (20), we get extension of gauss, confluent beta functions and their integral representations respectively which was defined by Özergin et al. [25]. For some p > 0, Mellin transform introduced by Mellin in 1897 (see [35]) as M [f(s); p] = F (p) = ∞∫ 0 sp−1f(s)ds. (22) The Laplace trasformation of f(v) for ℜ(s) > 0 is defined (see [35]) as F (s) = L{f(v)} = ∫ ∞ 0 e−svf(v)dv. (23) The Riemann-Liouville k-fractional integral of order −µ is defined as follows (see [3]) kI −µ v (f(v)) = 1 kΓk(−µ) ∫ v 0 (v − t) −µ k −1f(t)dt, ( k ∈ R+;R(µ) < 0 ) . (24) Remark 2. If we take k = 1 into (24), then k-Riemann-Liouville fractional integral of order −µ is reduced to Riemann-Liouville fractional integral of order −µ which defined in [36]. S. A. H. Shah et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6279 7 of 23 The classical Whitttaker function introduced by Whittaker [29] is Mτ,p(z) = zp+1/2 exp ( −z 2 ) ϕ ( p− τ + 1 2 ; 2p+ 1; z ) , (25) where ℜ(p) > −1 2 ;ℜ(p± τ) > −1 2 . Nagar et al. [30], investigated the extended whittaker function as follows Mλ,τ,p(z) = zp+1/2 exp ( −z 2 ) ϕλ ( p− τ + 1 2 : 2p+ 1 : z ) , (26) where λ ≥ 0,ℜ(p) > −1 2 ;ℜ(p± τ) > −1 2 . The generalized extended Whittaker function is defined as in [1] M (δ1,δ2,l1:l2) ξ,p,µ (z) = zµ+ 1 2 exp ( −z 2 ) ϕ (δ1,δ2,l1:l2) ξ (µ− p+ 1 2 : 2µ+ 1 : z), (27) where l1, l2 ≥ 1, ξ ∈ R+ 0 ,ℜ(µ) > −1 2 ,ℜ(µ± p) > −1/2, z ∈ C|(−∞,0],ℜ(δ1) > 0,ℜ(δ2) > 0. In [34], Savita Panwar and Prakriti Rai introduced Whittaker k-function as Mτ,p,k(z) = zp+1/2 exp ( −z 2 ) 1F1,k ( p− τ + 1 2 : 2p+ 1 : z ) , (28) where ℜ(p) > −1 2 ; ℜ(p± τ) > −1 2 , z ∈ C|(−∞,0]. Remark 3. If we take k=1 into (28) then (28), reduces to (25). By keeping in view the direction of the researchers in the field of special functions, we generalize some known functions like confluent hypergeometric, and Whittaker functions, prove some of their related properties as follows: 2. Generalized Extended Confluent Hypergeometric And Beta k-Function In this section, first we generalize the beta function in terms of new parameter k > 0, then we use definition of these generalized extended beta k-function to generalize the con- fluent hypergeometric in term of k > 0. We also investigate some properties like as integral representation, Mellin transforms, inverse Mellin transforms, Laplace transformation and derivative of these new generalized extended confluent hypergeometric k-functions. Definition 1. If k > 0, min{ℜ(δ1),ℜ(δ2),ℜ(l1),ℜ(l2)} > 0,ℜ(ξ) ≥ 0,ℜ(p),ℜ(q) > 0, l1, l2 ≥ 1, then we define the generalized extended beta k-function as β (δ1,δ2,l1,l2) ξ,k (p, q) = 1 k 1∫ 0 s p k −1(1− s) q k −1 S. A. H. Shah et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6279 8 of 23 ×1F1,k(δ1; δ2; −ξk ksl1(1− s)l2 )ds. (29) Definition 2. By using above definition (29), we extend the confluent hypergeometric function in terms of new parameter k > 0 as follows: If min{ℜ(δ1),ℜ(δ2),ℜ(l1),ℜ(l2)} > 0,ℜ(ξ) ≥ 0,ℜ(p),ℜ(q) > 0, l1, l2 ≥ 1, then ψ (δ1,δ2,l1:l2) ξ,k (σ2, σ3; t) = ∞∑ m=0 β (δ1,δ2,l1,l2) ξ,k (σ2 +mk, σ3 − σ2) βk(σ2, σ3 − σ2) tm m! . (30) Remark 4. If we take k = 1, then (29), (30) reduces to generalize extended beta and confluent hypergeometric functions which defined by Khan et al. [1]. If we take k = 1 and l1 = l2 into (29), (30), we get extended beta and confluent hypergeometric functions which introduced by Parmar [24]. If we take σ1 = σ2 and l1 = l2 = 1 and k = 1 into (29), (30), we get extended beta and confluent hypergeometric functions which introduced by Chaudhry et al. [22, 23]. Further if we take k=1 and l1 = l2 = 1 into (29), (30), we get extension of beta and confluent hypergeometric functions which defined by Özergin et al. [25]. 3. Integral Representations of Generalized Extended Confluent Hypergeometric k-Function Theorem 1. If k > 0, min{ℜ(δ1),ℜ(δ2),ℜ(l1),ℜ(l2)} > 0, ω ≥ 0,ℜ(σ3) > ℜ(σ2) > 0, then following integral representations hold true ψ (δ1,δ2,l1:l2) ξ,k (σ2, σ3; z) = 1 kβk(σ2, σ3 − σ2) 1∫ 0 s σ2 k −1(1− s) σ3−σ2 k −1 exp(zs) ×1F1,k(δ1; δ2; −ξk ksl1(1− s)l2 )ds (31) ψ (δ1,δ2,l1:l2) ξ,k (σ2, σ3; z) = 1 kβk(σ2, σ3 − σ2) ∞∫ 0 (u) σ2 k −1 (1 + u) σ3 k exp( zu 1 + u ) ×1F1,k(δ1; δ2; −ξk(1 + u)l1+l2 kul1 )ds (32) ψ (δ1,δ2,l1:l2) ξ,k (σ2, σ3; z) = 2 kβk(σ2, σ3 − σ2) π 2∫ 0 cos 2σ2 k −1 θ sin 2(σ3−σ2) k −1 θ exp(z cos2 θ) ×1F1,k(δ1; δ2; −ξk sec2l1 θ csc2l2 θ k )dθ (33) ψ (δ1,δ2,l1:l2) ξ,k (σ2, σ3; z) = 2 kβk(σ2, σ3 − σ2) π 4∫ 0 tanh 2σ2 k −1 θsech 2(σ3−σ2) k θ exp(z tanh2 θ) S. A. H. Shah et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6279 9 of 23 ×1F1,k(δ1; δ2; −ξk coth2l1 θ cosh2l2 θ k )dθ (34) ψ (δ1,δ2,l1:l2) ξ,k (σ2, σ3; z) = (q − p)1− σ3 k kβk(σ2, σ3 − σ2) q∫ p (u− p) σ2 k −1(q − u) σ3−σ2 k −1 exp [ z(u− p) q − p ] ×1F1,k(δ1; δ2; −ξk(q − p)l1+l2 k(u− p)l1(q − u)l2 )du (35) ψ (δ1,δ2,l1:l2) ξ,k (σ2, σ3; z) = 21− σ3 k kβk(σ2, σ3 − σ2) 1∫ −1 (1 + u) σ2 k −1(1− u) σ3−σ2 k −1 exp [ z(u+ 1) 2 ] ×1F1,k(δ1; δ2; −ξk(2)l1+l2 k(u+ 1)l1(1− u)l2 )du. (36) Proof. From equation (29), we have β (δ1,δ2,l1,l2) ξ,k (r2 +mk, r3 − r2) = 1 k 1∫ 0 s r2 k +m−1(1− s) r3−r2 k −1 ×1F1,k(δ1; δ2; −ξk ksl1(1− s)l2 )ds. (37) By using equation (37) into (30) and by changing the order of integration and summation, we get ψ (δ1,δ2,l1:l2) ξ,k (σ2, σ3; z) = ∞∑ m=0 kβk(σ2, σ3 − σ2) 1∫ 0 s r2 k +m−1(1− s) r3−r2 k −1 ×1F1,k(δ1; δ2; −ξk ksl1(1− s)l2 ) zm m! ds = 1 kβk(σ2, σ3 − σ2) 1∫ 0 s σ2 k −1(1− s) σ3−σ2 k −1 ∞∑ m=0 (zs)m m! ×1F1,k(δ1; δ2; −ξk ksl1(1− s)l2 )ds = 1 kβk(σ2, σ3 − σ2) 1∫ 0 s σ2 k −1(1 + s) σ3−σ2 k −1 exp(zs) ×1F1,k(δ1; δ2; −ξk ksl1(1− s)l2 )ds. which is (31) further by putting s = u 1+u , s = cos2 θ, s = tanh2 θ, s = u−p q−p and s = u+1 2 into equation (31), we get equations (32), (33), (34), (35), and (36) S. A. H. Shah et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6279 10 of 23 Remark 5. If we put k = 1 into equations (31), (32), (33), (34), (35), (36), we get integral representations of generalized extended confluent hypergeometric functions. 4. Mellin Transform And Transformation Formula of Generalized Extended Confluent Hypergeometric k-Function Theorem 2. If k > 0, ℜ(r) > 0, ℜ(δ1 + r) > 0, ℜ(δ2 + r) > 0, ℜ(ξ) ≥ 0, ℜ(δ1) > 0, ℜ(δ2) > 0, l1, l2 ≥ 1, then following Mellin tranformations holds true ∞∫ 0 ξr−1ψ (δ1,δ2,l1:l2) ξ,k (σ2, σ3; z)dξ = Γ (δ1,δ2) k (λ)βk(σ2 + l1r, σ3 − σ2 + l2r) βk(σ2, σ3 − σ2) ×1F1,k(σ2 + l1r, σ3 + (l1 + l2)r; z). (38) Proof. Multiplying equation (30) by ξr−1 on both sides and integrate w.r.t ξ from ξ = 0 to ξ = ∞ and by changing the order of integraton and summation, we get ∞∫ 0 ξr−1ψ (δ1,δ2,l1:l2) ξ,k (σ2, σ3; z)dξ = 1 kβk(σ2, σ3 − σ3) 1∫ 0 s σ2 k −1(1− s) σ3−σ2 k −1 exp(zs)ds × ∞∫ 0 ξr−1 1F1,k(δ1; δ2; −ξk ksl1(1− s)l2 )dξ. (39) By substituting λ = ξ s l1 k (1−s) l2 k into (39), we get ∞∫ 0 ξr−1 1F1,k(δ1; δ2; −ξk ksl1(1− s)l2 )dξ = s rm1 k (1− s) n1r k ∞∫ 0 λr−1 1F1,k(σ1, σ2; −λk k )dλ = s l1r k (1− s) n1r k Γ (σ1,σ2) k (λ). (40) By substituting equation (40) into equation (39), we get = Γ (p1,q1) k (λ) kβk(σ2, σ3 − σ3) 1∫ 0 s σ2+rl1 k −1(1− s) (σ3−σ2)+rl2 k −1 exp(zs)ds = Γ (δ1,δ2) k (λ)βk(σ2 + l1r, σ3 − σ2 + l2r) βk(σ2, σ3 − σ2) 1F1,k(σ2 + l1r, σ3 + (l1 + l2)r; z). Corollary 1. By the Mellin inversion formula, we have following integral of genetalized extended confluent hypergeometric k-function ψ (δ1,δ2,l1:l2) ξ,k (σ2, σ3; z) = 1 2πι c+ι∞∫ c−ι∞ Γ (δ1,δ2) k (λ)βk(σ2 + l1r, σ3 − σ2 + l2r) βk(σ2, σ3 − σ2) S. A. H. Shah et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6279 11 of 23 ×1F1,k(σ2 + l1r, σ3 + (l1 + l2)r; z)ξ −rdξ. (41) Proof. By taking Mellin inverse of equation (38) on both sides, we get the required result. Remark 6. If we take k = 1 into equation (38), (41), then we get Mellin and inverse Mellin transforms of generalized extended confluent hypergeometric function which intro- duced by Khan et al. [1]. Theorem 3. If k > 0, min{ℜ(δ1),ℜ(δ2),ℜ(l1),ℜ(l2)} > 0,ℜ(ξ) ≥ 0,ℜ(δ2),ℜ(δ3) > 0, l1, l2 ≥ 1, then following transformation formula holds for generalized extended confluent hypergeometric k-function ψ (δ1,δ2,l1:l2) ξ,k (σ2, σ3; t) = exp(t)ψ (δ1,δ2,l1:l2) ξ,k (σ3 − σ2, σ3;−t). (42) Proof. By using integral representation of generalized extended confluent hypergeo- metric k-function, we have ψ (δ1,δ2,l1:l2) ξ,k (σ2, σ3; z) = 1 kβk(σ2, σ3 − σ2) 1∫ 0 s σ2 k −1(1− s) σ3−σ2 k −1 exp(zs) ×1F1,k(δ1; δ2; −ξk ksl1(1− s)l2 )ds. Replace s by s− 1, we get the required result. Remark 7. If we put k = 1 into equation (42), we get generalized extended Kummar’s first fomula defined by Khan et al. [1]. Further if we take m=n, then (42) reduces to extended Kummar’s first fomula defined by Parmar in [24]. If we take ξ = 0, then we get classical Kummar’s first fomula. 5. Laplace Transformation of Generalized Extended Confluent Hypergeometric k-Function Theorem 4. If k > 0,min{ℜ(δ1),ℜ(δ2),ℜ(l1),ℜ(l2)} > 0, ω ≥ 0,ℜ(σ3) > ℜ(σ2) > 0,ℜ(d) > 0, then ∞∫ 0 e−sttd−1ψ (δ1,δ2,l1:l2) ξ,k (σ2, σ3; vt)dt = ∞∑ m=0 Γ(m+ d) sm+d × β (δ1,δ2,l1,l2) ξ,k (σ2 +mk, σ3 − σ2) βk(σ2, σ3 − σ2) vm m! . (43) S. A. H. Shah et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6279 12 of 23 Proof. Consider left hand side of equation (43), we have ∞∫ 0 e−sttd−1ψ (δ1,δ2,l1:l2) ξ,k (σ2, σ3; vt)dt = ∞∫ 0 e−sttd−1 ∞∑ m=0 β (δ1,δ2,l1,l2) ξ,k (σ2 +mk, σ3 − σ2) βk(σ2, σ3 − σ2) ×v mtm m! dt. By changing the order of integration and summation, we have ∞∫ 0 e−sttd−1ψ (δ1,δ2,l1:l2) ξ,k (σ2, σ3; vt)dt = ∞∑ m=0 β (δ1,δ2,l1,l2) ξ,k (σ2 +mk, σ3 − σ2) βk(σ2, σ3 − σ2) vm m! ∞∫ 0 e−sttm+d−1dt = ∞∑ m=0 β (δ1,δ2,l1,l2) ξ,k (σ2 +mk, σ3 − σ2) βk(σ2, σ3 − σ2) vm m! Γ(m+ d) sm+d = ∞∑ m=0 Γ(m+ d) sm+d β (δ1,δ2,l1,l2) ξ,k (σ2 +mk, σ3 − σ2) βk(σ2, σ3 − σ2) vm m! . Theorem 5. If k > 0,min{ℜ(δ1),ℜ(δ2),ℜ(l1),ℜ(l2)} > 0, ω ≥ 0,ℜ(σ3) > ℜ(σ2) > 0,ℜ(d) > 0, then ∞∫ 0 e−stt d k −1ψ (δ1,δ2,l1:l2) ξ,k (σ2, σ3; vt)dt = ∞∑ m=0 (σ1)m,kβ (δ1,δ2,l1,l2) ξ,k (σ2 +mk, σ3 − σ2)Γk(d+mk) βk(σ2, σ3 − σ2)k d k +m−1s d k +m ×v m m! . (44) Proof. Consider left hand side of equation (44), we have ∞∫ 0 e−stt d k −1ψ (δ1,δ2,l1:l2) ξ,k (σ2, σ3; vt)dt = ∞∫ 0 e−stt d k −1 ∞∑ m=0 β (δ1,δ2,l1,l2) ξ,k (σ2 +mk, σ3 − σ2) βk(σ2, σ3 − σ2) vmtm m! dt. By changing the order of integration and summation, we have ∞∫ 0 e−stt d k −1ψ (δ1,δ2,l1:l2) ξ,k (σ2, σ3; vt)dt = ∞∑ m=0 β (δ1,δ2,l1,l2) ξ,k (σ2 +mk, σ3 − σ2) βk(σ2, σ3 − σ2) vm m! ∞∫ 0 e−stt d k +m−1dt = ∞∑ m=0 β (δ1,δ2,l1,l2) ξ,k (σ2 +mk, σ3 − σ2)Γk( d k +m)k βk(σ2, σ3 − σ2)k d k +m−1s d k +m vm m! . S. A. H. Shah et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6279 13 of 23 6. Derivative of Generalized Extended Confluent Hypergeometric k-Function Theorem 6. If k > 0,min{ℜ(δ1),ℜ(δ2),ℜ(l1),ℜ(l2)} > 0, ω ≥ 0,ℜ(σ3) > ℜ(σ2) > 0, l1, l2 ≥ 1, then dl dvl [ψ (δ1,δ2,l1:l2) ξ,k (σ2, σ3; v)] = (σ2)l,k (σ3)l,k ψ (δ1,δ2,l1:l2) ξ,k (σ2 + lk, σ3 + lk; v). (45) Proof. From equation (30), we have ψ (δ1,δ2,l1:l2) ξ,k (σ2, σ3; v) = ∞∑ l=0 β (δ1,δ2,l1,l2) ξ,k (σ2 + lk, σ3 − σ2) βk(σ2, σ3 − σ2) vl l! . (46) After taking derivative of (46) w.r.t v and simlification, we obtain d dv [ψ (δ1,δ2,l1:l2) ξ,k (σ2, σ3; v)] = ∞∑ l=0 β (δ1,δ2,l1,l2) ξ,k (σ2 + lk, σ3 − σ2) βk(σ2, σ3 − σ2) lvl−1 l! = ∞∑ l=1 β (δ1,δ2,l1,l2) ξ,k (σ2 + lk, σ3 − σ2) βk(σ2, σ3 − σ2) vl−1 (l − 1)! . By replacing l by l+1, we get d dv [ψ (δ1,δ2,l1:l2) ξ,k (σ2, σ3; v)] = ∞∑ l=0 β (δ1,δ2,l1,l2) ξ,k (σ2 + (l + 1)k, σ3 − σ2) βk(σ2, σ3 − σ2) vl l! (47) By using following property βk(σ2, σ3 − σ2) = (σ3)k (σ2)k βk(σ2 + k, σ3 − σ2), then equation (47) can be written as d dv [ψ (δ1,δ2,l1:l2) ξ,k (σ2, σ3; v)] = (σ2)k (σ3)k ∞∑ l=0 β (δ1,δ2,l1,l2) ξ,k (σ2 + (l + 1)k, σ3 − σ2) βk(σ2 + k, σ3 − σ2) vl l! . Suppose result is true for l-1, then dl−1 dvl−1 [ψ (δ1,δ2,l1:l2) ξ,k (σ2, σ3; v)] = (σ2, )l−1,k (σ3)l−1,k (48) S. A. H. Shah et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6279 14 of 23 ×ψ(δ1,δ2,l1:l2) ξ,k (σ2 + (l − 1)k, σ3 + (l − 1)k; v). By taking derivative of equation (48) w.r.t. v, we get dl dvl [ψ (δ1,δ2,l1:l2) ξ,k (σ2, σ3; v)] = (σ2)l−1,k(σ2 + (l − 1)k) (σ3)l−1,k(σ3 + (l − 1)k) ×ψ(δ1,δ2,l1:l2) ξ,k (σ1 + lk, σ2 + lk, σ3 + lk; v) = (σ2)l,k (σ3)l,k ψ (δ1,δ2,l1:l2) ξ,k (σ2 + lk, σ3 + lk; v). Remark 8. If we take k = 1 into equation (45), we get derivative of generalized extended confluent hypergeometric function which investigated by Khan et al. [1]. 7. Generalized Extended Whittaker k-Function In this section, we introduce generalized extended whittaker k-function with the help of generalized extended confluent hypergeometric k-function. Further, we investigate Mellin transforms, Hankel transformation, Laplace transformation, fractional integral and deriva- tive of these new generalized extended Whittaker k-function. Definition 3. If ξ ≥ 0, l1, l2 ≥ 1,ℜ(δ1),ℜ(δ2) > 0, k > 0,ℜ(µ) > −1 2 ,ℜ(µ + p) > −1 2 ,ℜ(µ− p) > −1 2 ,then generalized extended Whittaker k-function we define as follows M (δ1,δ2,l1:l2) ξ,k,p,µ (z) = zµ+ 1 2 exp ( −z 2 ) ψ (δ1,δ2,l1:l2) ξ,k (µ− p+ 1 2 ; 2µ+ 1; z). (49) where ψ (δ1,δ2,l1:l2) ξ,k (u, v; z) is generalized extended confluent hypergeometric k-function which is defined in (30). Remark 9. If we take k = 1 into (49), we get generalized extended Whittaker function defined by Khan et al. [1]. If we take l2 = 1, then we get extended Whittaker function defined in [32],also see[33]. Further by putting δ1 = δ2, and l1 = l2, we get extended Whittaker function which investigated by Khan and Ghayasuddin in [31]. Further for l2 = 1, we get extended Whittaker due to Nagar et al. [30]. For ξ = 0 gives classical Whittaker function defined in [29] 8. Integral Representation of Generalized Extended Whittaker k-Function Theorem 7. If ξ ≥ 0,ℜ(µ+ p) > −1 2 ,ℜ(µ− p) > −1 2), l1, l2 ≥ 1,ℜ(δ1),ℜ(δ2) > 0, k > 0, then following integral representations hold true M (δ1,δ2,l1:l2) ξ,k,p,µ (z) = zµ+ 1 2 exp(−z 2 ) kβk(µ− p+ 1 2 , µ+ p+ 1 2) 1∫ 0 t µ−p+1 2 k −1(1− t) µ+p+1 2 k −1 exp(zt) S. A. H. Shah et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6279 15 of 23 ×1F1,k(δ1; δ2; −ξk ktl1(1− t)l2 )dt. (50) M (δ1,δ2,l1:l2) ξ,k,p,µ (z) = (q − s)1− 2µ+1 k zµ+ 1 2 exp(−z 2 ) kβk(µ− p+ 1 2 , µ+ p+ 1 2) q∫ s (u− s) µ−p+1 2 k −1(q − u) µ+p+1 2 k −1 × exp [ z(u− s) q − s ] 1F1,k(δ1; δ2; −(q − s)l1+l2ξk k(u− s)l1(q − u)l2 )du. (51) M (δ1,δ2,l1:l2) ξ,k,p,µ (z) = zµ+ 1 2 exp(−z 2 ) kβk(µ− p+ 1 2 , µ+ p+ 1 2) ∞∫ 0 u µ−p+1 2 k −1(u+ 1) −(2µ+1) k × exp( zu 1 + u )1F1,k(p1, q1; −ξk(u+ 1)l1+l2 kul1 )du. (52) M (δ1,δ2,l1:l2) ξ,k,p,µ (z) = (2)1− 2µ+1 k zµ+ 1 2 kβk(µ− p+ 1 2 , µ+ p+ 1 2) 1∫ −1 (u+ 1) µ−p+1 2 k −1(1− u) µ+p+1 2 k −1 × exp( zu 2 )1F1,k(δ1; δ2; −(2)l1+l2ξk k(u+ 1)l1(1− u)l2 )du. (53) M (δ1,δ2,l1:l2) ξ,k,p,µ (z) = zµ+ 1 2 exp( z2) kβk(µ− p+ 1 2 , µ+ p+ 1 2) 1∫ 0 (1− u) µ−p+1 2 k −1(u) µ+p+1 2 k −1 exp(−zu) ×1F1,k(δ1; δ2; −ξk k(1− u)l1(u)l2 )du. (54) Proof. By using equation (31) in (49), we get (50) and further by putting t = u−s q−s , t = u 1+u , t = 1− u, in (50), we get (51), (52), (54). If we take s = −1 and q = 1 in (51), we get (53) Remark 10. If we take k = 1 in (50), (51), (52), (53), and (54), we get integral rep- resentation of generalized extended Whittaker function investigated by Khan et al. [1]. Further by putting δ1 = δ2, and l1 = l2 we get integral representation of extended Whit- taker function which investigated by Khan and Ghayasuddin in [31], further for l2 = 1, we get integral representation of extended whittaker due to Nagar et al. [30]. S. A. H. Shah et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6279 16 of 23 9. Integral Transform of Generalized Extended Whittaker k-Function Theorem 8. If k > 0, ℜ(r) > 0, ℜ(δ1 + r) > 0, ℜ(δ2 + r) > 0, ℜ(ξ) ≥ 0, ℜ(δ1) > 0, ℜ(δ2) > 0, l1, l2 ≥ 1, ℜ(µ + l2r + k) > −1 2 , R(µ + l1r − k) > −1 2 , then following Mellin transforms holds true ∞∫ 0 ξr−1M (δ1,δ2,l1,l2) ξ,k,p,µ (z)dξ = Γ (δ1,δ2) k (r)zµ+ 1 2 exp(−z 2 )βk(µ− p+ 1 2 + l1r, µ+ p+ 1 2 + l2r) βk(µ− p+ 1 2 , µ+ p+ 1 2) ×ψk(µ− p+ 1 2 + l1r, 2µ+ (l1 + l2)r + 1). (55) Proof. Consider left hand side of equation (55) then using (50) and changing the order of integration, we get ∞∫ 0 ξr−1M (δ1,δ2,l1:l2) ξ,k,p,µ (z)dξ = zµ+ 1 2 exp(−z 2 ) kβk(µ− p+ 1 2 , µ+ p+ 1 2) 1∫ 0 s µ−p+1 2 k −1(1− s) µ+p+1 2 k −1 exp(zs) × ∞∫ 0 ξr−1 1F1,k(δ1; δ2; −ξk ksl1(1− s)l2 )dξ. By substituting λ = ξ s m1 k (1−s) n1 k , we get ∞∫ 0 ξr−1M (δ1,δ2,l1:l2) ξ,k,p,µ (z)dξ = Γ (p1,q1) k (r)zµ+ 1 2 exp(−z 2 ) kβk(µ− p+ 1 2 , µ+ p+ 1 2) 1∫ 0 s µ−p+1 2+m1r k −1(1− s) µ+p+1 2+n1r k −1 exp(zs)ds = Γ (p1,q1) k (r)zµ+ 1 2 exp(−z 2 ) kβk(µ− p+ 1 2 , µ+ p+ 1 2) βk(µ− p+ 1 2 +m1r, µ+ p+ 1 2 + n1r) βk(µ− p+ 1 2 +m1r, µ+ p+ 1 2 + n1r) × 1∫ 0 s µ−p+1 2+m1r k −1(1− s) µ+p+1 2+n1r k −1 exp(zs)ds. By using the integral representation of generalized extended confluent hypergeometric k-function we get the above result. Remark 11. If we take k = 1 in (55), then we get Mellin transfrmation of generalized extended confluent hypergeometric function which was investigated by Khan et al. [1]. Theorem 9. If ξ ≥ 0, 2p > b > 0, ℜ(a+µ) > −1 2 , k > 0, then we have following integral transfom holds true ∞∫ 0 exp(−pz)za−1M (δ1,δ2,l1:l2) ξ,k,p,µ (bz)dz = (b)µ+ 1 2Γ(a+ µ+ 1 2) (p+ b 2) a+µ+ 1 2 S. A. H. Shah et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6279 17 of 23 ×F δ1,δ2,l1,l2 ξ,k (a+ µ+ 1 2 , µ− p+ 1 2 ; 2µ+ 1; 2b 2p+ b ). (56) Proof. Consider the left hand side of equation (56), then using integral representation of generalized extended Whittaker k-function and by changing the order of interation and summation, we get ∞∫ 0 exp(−pz)za−1M (δ1,δ2,l1:l2) ξ,k,p,µ (bz)dz = ∞∑ m=0 β (δ1,δ2,l1,l2) ξ,k (µ− p+ 1 2 +mk, µ+ p+ 1 2) βk(µ− p+ 1 2 , µ+ p+ 1 2) ×(b)m+µ+ 1 2 m! ∞∫ 0 exp(−(p+ b 2 )z)z(a+µ+m+ 1 2 )−1dz. By using the following formula Γ(u) su = ∞∫ 0 exp(−st)tu−1dt, we get the above result. Corollary 2. If we put a=b=1 in (56), then we get following integral transform ∞∫ 0 exp(−pz)M (δ1,δ2,l1:l2) ξ,k,p,µ (z)dz = (2)µ+ 3 2Γ(µ+ 3 2) (2p+ 1)µ+ 3 2 ×F δ1,δ2,l1,l2 ξ,k (µ+ 3 2 , µ− p+ 1 2 ; 2µ+ 1; 2 2p+ 1 ). Theorem 10. If k > 0, ℜ(µ + p) > −1 2 , ℜ(µ − p) > −1 2 , ℜ(µ + v) > 0, l1, l2 ≥ 1, then following Hankel transformation holds true ∞∫ 0 zM (δ1,δ2,l1:l2) ξ,k,p,µ (z)Jv(az)dz = Γ(µ+ v + 5 2) (a2 + 1 4) µ 2 + 5 4 ∞∑ m=0 β (δ1,δ2,l1,l2) ξ,k (µ− p+ 1 2 +mk, µ+ p+ 1 2) βk(µ− p+ 1 2 , µ+ p+ 1 2) × Γ(µ+ v + 5 2)m (a2 + 1 4) m 2 m! P−v µ+m+ 3 2 ( 1√ 4a2 + 1 ). (57) Proof. By using (26) and (46), then by changing the order of integration and summa- tion, we get ∞∫ 0 zM (δ1,δ2,l1:l2) ξ,k,p,µ (z)Jv(az)dz = ∞∑ m=0 β (δ1,δ2,l1,l2) ξ,k (µ− p+ 1 2 +mk, µ+ p+ 1 2) βk(µ− p+ 1 2 , µ+ p+ 1 2)m! S. A. H. Shah et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6279 18 of 23 × ∞∫ 0 z(µ+m+ 3 2 ) exp( −z 2 )Jv(az)dz. By using the following formula ∞∫ 0 exp(−pz)zµJv(az)dz = Γ(µ+ v + 1)r−µ−1P−v µ ( p r ). R(µ+ v) > −1, r = √ p2 + a2, P−v µ (z) is Legendre function. By taking p = 1 2 , µ = µ+m+ 3 2 and after simlification, we get requir result. Theorem 11. For generalzed extended Whittaker k-function the following relation holds true M (δ1,δ2,l1:l2) ξ,p,µ,k (−z) = (−1)µ+ 1 2M (δ1,δ2,l1:l2) ξ,k,−p,µ (z), (58) where ξ ≥ 0,ℜ(µ) > −1 2 ,ℜ(µ+ p) > −1 2 ,ℜ(µ− p) > −1 2 , l1, l2 ≥ 1. Proof. By replacing z by −z in equation (49), we get M (δ1,δ2,l1:l2) ξ,k,p,µ (−z) = (−z)µ+ 1 2 exp (z 2 ) ψ (δ1,δ2,l1:l2) ξ,k (µ− p+ 1 2 , 2µ+ 1;−z), (59) Now using (30) in (59) and after simplication, we get desired result. 10. Laplace Transformation of Generalized Extended Whittaker k-Function Theorem 12. If k > 0, ℜ(ξ + µ) > 1 2 , ℜ(ξ − µ) > 1 2 , l1, l2 ≥ 1, then L[exp( z 2 )M (δ1,δ2,l1:l2) ξ,k,p,µ (z)] = ∞∑ m=0 Γ(µ+m+ 3 2) (s)µ+m+ 3 2 × β (δ1,δ2,l1,l2) ξ,k (µ− p+ 1 2 +mk, µ+ p+ 1 2) βk(µ− p+ 1 2 , µ+ p+ 1 2) m! . (60) Proof. By using equation (49) and definition of Laplace transform, we get exp( z 2 )M (δ1,δ2,l1:l2) ξ,k,p,µ (z) = ∞∫ 0 exp(−sz)zµ+ 1 2ψ (δ1,δ2,l1,l2) ξ,k (µ− p+ 1 2 ; 2µ+ 1; z)dz. By using (26) and by changing the order of integration and summation, we have = ∞∑ m=0 β (δ1,δ2,l1,l2) ξ,k (µ− p+ 1 2 +mk, µ+ p+ 1 2) βk(µ− p+ 1 2 , µ+ p+ 1 2)m! S. A. H. Shah et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6279 19 of 23 × ∞∫ 0 exp(−sz)z(µ+m+ 3 2 )−1dz, By using the definition of classical gamma function, we get desired result. 11. Riemann-Liouville Fractional Integral of Generalized Extended Whittaker k-Function Theorem 13. If ℜ(λ) < 0, ξ ∈ R+ 0 , ℜ(µ + p) > −1 2 , ℜ(µ − p) > −1 2 , k > 0, l1, l2 ≥ 1, then ℶλ z [exp( z 2 )M (δ1,δ2,l1:l2) ξ,k,p,µ (z)] = zµ−λ+ 1 2 Γ(−λ) ∞∑ l=0 β (δ1,δ2,l1,l2) ξ,k (µ− p+ 1 2 + lk, µ+ p+ 1 2) βk(µ− p+ 1 2 , µ+ p+ 1 2) zl l! ×β(µ+ l + 3 2 ;−λ). (61) Proof. By using definition of Riemann-Liouville fractional integral, we get ℶλ z [exp( z 2 )M (δ1,δ2,l1:l2) ξ,k,p,µ (z)] = 1 Γ(−λ) z∫ 0 t 1 2 +µψ (δ1,δ2,l1:l2) ξ,k (µ− p+ 1 2 , 2µ+ 1; t)(z − t)−λ−1dt. Now using equation (49), then use (31) and by changing the order of integration and summation, we have ℶλ z [exp( z 2 )M (δ1,δ2,l1:l2) ξ,k,p,µ (z)] = ∞∑ l=0 β (δ1,δ2,l1,l2) ξ,k (µ− p+ 1 2 + lk, µ+ p+ 1 2) βk(µ− p+ 1 2 , µ+ p+ 1 2)l! × 1 Γ(−λ) z∫ 0 tµ+l+ 1 2 (z − t)−λ−1dt. By substituting t = vz, we get ℶλ z [exp( z 2 )M (δ1,δ2,l1:l2) ξ,k,p,µ (z)] = zµ−λ+ 1 2 Γ(−λ) 1∫ 0 vµ+l+ 1 2 (1− v)−λ−1dv × ∞∑ l=0 β (δ1,δ2,l1,l2) ξ,k (µ− p+ 1 2 + lk, µ+ p+ 1 2) βk(µ− p+ 1 2 , µ+ p+ 1 2) zl l! , By using definition of beta function, we get desired result. S. A. H. Shah et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6279 20 of 23 12. Riemann-Liouville k-fractional Integral of Generalized Extended Whittaker k-Function Theorem 14. If k > 0, ℜ(λ) < 0, ξ ∈ R+ 0 , ℜ(µ + p) > −1 2 , ℜ(µ − p) > −1 2 , l1, l2 ≥ 1, then kℶλ z [exp( z 2 )M (δ1,δ2,l1:l2) ξ,k,p,µ (z)] = zµ− λ k + 1 2 kΓk(−λ) ∞∑ l=0 β (δ1,δ2,l1,l2) ξ,k (µ− p+ 1 2 + lk, µ+ p+ 1 2) βk(µ− p+ 1 2 , µ+ p+ 1 2) zl l! ×β(µ+ l + 3 2 ; −λ k ). (62) Proof. By using definition of Riemann-Liouville k-fractional integral, we get kℶλ z [exp( z 2 )M (δ1,δ2,l1:l2) ξ,p,µ,k (z)] = 1 kΓk(−λ) z∫ 0 t 1 2 +µψ (δ1,δ2,l1:l2) ξ,k (µ− p+ 1 2 , 2µ+ 1; t)(z − t) −λ k −1dt. First use equation (49), then using (31) and by changing the order of integration and summation, we have kℶλ z [exp( z 2 )M (δ1,δ2,l1:l2) ξ,k,p,µ (z)] = ∞∑ l=0 β (δ1,δ2,l1,l2) ξ,k (µ− p+ 1 2 + lk, µ+ p+ 1 2) βk(µ− p+ 1 2 , µ+ p+ 1 2)l! . × 1 kΓk(−λ) z∫ 0 tµ+l+ 1 2 (z − t) −λ k −1dt. By substituting t = vz, we get kℶλ z [exp( z 2 )M (δ1,δ2,l1:l2) ξ,k,p,µ (z)] = zµ− λ k + 1 2 kΓk(−λ) 1∫ 0 vµ+l+ 1 2 (1− v) −λ k −1dv × ∞∑ l=0 β (δ1,δ2,l1,l2) ξ,k (µ− p+ 1 2 + lk, µ+ p+ 1 2) βk(µ− p+ 1 2 , µ+ p+ 1 2) zl l! . By using definition of beta k-function, we get desired result. 13. Derivative of Generalized Extended Whittaker k-Function Theorem 15. If ξ ≥ 0,ℜ(µ+ p) > −1 2 ,ℜ(µ− p) > −1 2), l1, l2 ≥ 1,ℜ(δ1),ℜ(δ2) > 0, k > 0, then dl dzl [exp( z 2 )z−µ− 1 2M (δ1,δ2,l1:l2) ξ,k,p,µ (z)] = (µ− p+ 1 2)l,k (2µ+ 1)l,k × exp( z 2 )z−µ− lk 2 − 1 2Mξ,p− lk 2 ,µ+ lk 2 ,k(z). (63) S. A. H. Shah et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6279 21 of 23 Proof. From equation (45), we have dl dvl [ψ (δ1,δ2,l1:l2) ξ,k (σ2, σ3; v)] = (σ2)l,k (σ3)l,k ψ (δ1,δ2,l1:l2) ξ,k (σ2 + lk, σ3 + lk; v). (64) M (δ1,δ2,l1:l2) ξ,p,µ,k (z) = (z)µ+ 1 2 exp ( −z 2 ) ×ψ(δ1,δ2,l1:l2) ξ,k (µ− p+ 1 2 , 2µ+ 1 : z). (65) Now consider left hand side of (63) and using definition of generalized extended Whit- taker k-function, we have dl dzl [exp( z 2 )z−µ− 1 2M (δ1,δ2,l1:l2) ξ,p,µ,k (z)] = dl dzl [ψ (δ1,δ2,l1:l2) ξ,k (µ− p+ 1 2 , 2µ+ 1+ : z)]. Now applying (64) in above equation and after simplification, we get desired result. 14. 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