Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2712 https://internationalpubls.com Computation of Euler-Type Integrals Involving Generalized M-Series M. U. Siddiqui 1, Owais Khan2*, Nafis Ahmad3 and M. Kashif Khan4 1,2,4Department of Mathematics and Statistics, Integral University, Lucknow-226026,India Emails: 2uzairmohd931@gmail.com, 2owkhan05@gmail.com, 4mdkashifkhan85@gmail.com 3Department of Mathematics, Shibli National College, Azamgarh-276001, India Email: 3nafis.sncmaths@gmail.com Article History: Received: 12-01-2025 Revised: 15-02-2025 Accepted: 01-03-2025 Abstract: : This research presents new evaluations of Euler-type integrals that encompass the generalized M-series, a comprehensive extension of special functions. These results not only deepen our understanding of the structural properties of the generalized M-series but also suggest potential applications in various fields, including mathematical physics, engineering, and applied mathematics. Several specific cases are analysed to illustrate the versatility and utility of the derived integral formulas by selecting particular parameter values of the generalized M-series. Keywords: : Euler-type integrals, generalized M-series and Fox-wright functions. 1. Introduction Special functions are mathematical functions that have been assigned specific names and symbols due to their importance in various fields, including functional analysis, geometry, physics, and mathematical analysis. Integral transforms are widely used in many applied mathematics and mathematical physics problems. Several mathematicians have developed integral transforms, such as the Euler integral, Laplace transform, Fourier transform, Mellin transform, and Hankel transform, each incorporating different special functions. In numerous studies, the term exp(t), which appears in the integral representation of the gamma function, is often substituted with more complex functions to facilitate generalizations. These extensions of the beta function are presented while maintaining its symmetry properties [1] . Furthermore, integral and derivative formulas for Gauss hypergeometric and confluent hypergeometric functions, along with descriptions of their generalizations, are derived using the generalized beta function. Recently, many researchers have been presenting extensions and generalizations for special functions, such as the Pochhammer symbol, gamma function, k-gamma function, p-k-gamma and beta functions, along with the confluent hypergeometric function (see [2-5, 9-10, 15]). Sharma and Jain [12] characterized a generalized M-series, which extends various special functions such as the Mittag-Leffler function, Wright function, Prabhaker function, and Gauss hypergeometric function. In 2018, Suthar et al. [15] assessed integral expressions of the product of M-series and Jacobi polynomials. Subsequently, Sachan et al. [11] presented a new extension of the M-series and discovered its properties, including recurrence relations, integral representation, and formulas for mailto:2 mailto:2owkhan05@gmail.com mailto:4mdkashifkhan85@gmail.com mailto:3nafis.sncmaths@gmail.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2713 https://internationalpubls.com fractional integrals and derivatives. Following this excellent work, we obtained composition formulas for Euler-type integrals featuring a generalized M-series kernel. Additionally, we demonstrated applications of our findings as specific instances by selecting appropriate values for the parameters of generalized M-series Sachan et al. [12] defined generalized M-series, which is novel an extension of generalized hypergeometric function, Wright function and Mittag-Leffler is defined as: Mq ρ p σ = Mq ρ (k1, … . , kp, l1 … . lq: z)p σ = ∑ (k1)m…(kp)m (l1)m…(lq)m ∞ m=0 zm ⌈(ρ+σm) , (1.1) where ρ, σ, z ϵ C, R(σ) > 0, (ki)m(i = 1, … . , p) and (lj)m (j = 1, … , q) . Well known Pochhamer symbols (k)m defined as: (k)m = { 1 m = 0, k ≠ 0 k(k + 1) … (k + m − 1), mϵN, kϵC = Γ(k+m) Γ(k) . The series (1.1) is convergent for all z if p ≤ q. it is convergent for |z| < η = σσ if p = q + 1 and divergent if p > q + 1. when p = ν + 1 and |z| = η, there is convergent under conditions that depend on parameters. The detailed accound of the M-series can be found in paper written by Sharma and Jain [12]. The generalized M-series can be represented as a special case of Wright generalized hypergeometric function, called Fox-Wright function rΨs[x], of the Fox H-function, and of Meijer G-function [6]. Mq ρ p σ (k1, … . , kp, l1 … . lq: z) = ∏ Γ(lq) q j=1 ∏ Γ(kp) p j=1 ∑ Γ(λ1+λ1k),……,Γ(λr+λsk)xk Γ(l1+l1k),…… ,Γ(l1+l1k)k! ∞ k=0 , (1.2) = p+1Ψq+1 [ (λ1, λ1), … … , (λr, λs); (l1, l1), … … ., (lr, ls); x] (1.3) Mq ρ p σ (k1, … . , kp, l1 … . lq: z) = ∏ Γ(lq) q j=1 ∏ Γ(kp) p j=1 Hp+1,q+2 1,p+1 [−z| (1 − kj)1 p , (0,1) (0,1), (1 − lj)1 q , (0,1) ], (1.4) where Hr,s+1 1,r [x] is a Fox-H function [1] and the coefficients γ1 ′ , … . , γ′r, l′1, … . , l′sϵ R+ such that 1 + ∑ lj ′s j=1 − ∑ γj ′r i=1 for suitable bounded value of |x|. Taking suitable values of the parameters in (1.1), we conclude special cases and connections which are enumerated as follows: 1. Forσ = 1, the generalized M-series reduces in the M-series defined by Sharma [12]. Mq ρ p 1 (k1, … . , kp, l1 … . lq: z) = ∑ (k1)m……(kp) m (l1)m…….(lq) m ∞ m=0 zm Γ(ρ+σm) (1.5) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2714 https://internationalpubls.com = pMq ρ (k1, … . , kp, l1 … . lq: z) 2. For p = q = 0, the generalized M-series reduces in the Mittag-Leffler function[17] M0 ρ 0 σ (k1, … . , kp, l1 … . lq: z) = ∑ zm Γ(ρ+σm) = Eρ,σ(z)∞ m=0 (1.6) 3. For p = q = 1, k = αϵ C, l = 1, equation (1.1) reduces in the generalized Mittag-Leffler function [14] M1 ρ 1 σ (−, −: z) = ∑ zm Γ(ρ+σm) = Eρ,σ α (z)∞ m=0 (1.7) 4. For p = q = 1, k = αϵ C, l = 1, equation (1.1) reduces in the Wright function [17] M1 ρ 1 1 (−,1: z) = ∑ zm Γ(ρ+m) = Wρ,σ(z)∞ m=0 (1.8) 5. For ρ = σ = 1, equation (1.1) reduces in term of Gauss hypergeometric function 𝑀𝑞 1 𝑝 1 (𝑘1, … . , 𝑘𝑝, 𝑙1 … . 𝑙𝑞: 𝑧) = ∑ (k1)m……(kp) m (l1)m…….(lq) m ∞ m=0 zm 𝑚! = = 𝑝F𝑞 [ k1, … . . kp; l1, … … . lq; z]. (1.9) In the present study, we also need to recall the following interesting and useful integral identities in term of gamma function [1, 14-15] as follows: 1. ∫ e−αt[sinh (βt)]γdt ∞ 0 = β−12−γ−1 Γ( α 2β − γ 2 )Γ(1+γ) Γ( 1 2β + γ 2 +1) , (1.10) a. 𝑅(𝛾) > −1, 𝑅(𝛽) > 0, 𝑅( 𝛼 𝛽 ) > 𝑅(𝛾) 2. ∫ xσ(1 − x2)− μ 2Pγ μ(x)dx = 2μ−1 Γ( 1 2 + σ 2 )Γ(1+ σ 2 ) Γ(1+ σ 2 − ν 2 − μ 2 )Γ( σ 2 + ν 2 − μ 2 + 3 2 ) 1 0 , (1.11) a. 𝑅𝑒(𝜇) < 1, 𝑅𝑒(𝜎) > −1 3. . ∫ 𝑥−𝜌(𝑥2 − 1)− 𝜇 2 ∞ 1 𝑃𝜈 𝜇(𝑥)𝑑𝑥 = 2𝜌+𝜇−2 𝛤( 𝜌+𝜇+𝜈 2 )𝛤( 𝜌+𝜇−𝜈−1 2 ) 𝜋 1 2𝛤(𝜌) , (1.12) 𝑅𝑒(𝜇) < 1, 𝑅𝑒(𝜌 + 𝜇 + 𝜈) > 0, 𝑅𝑒(𝜌 + 𝜇 − 𝜈) > 1 Integral containing Tchebichef polynomial 4. ∫ (1 − x) 1 2(1 + x)αUn(x)dx = 1 −1 π 1 22α+2μ+ 3 2 [n+1]2Γ(α+ 1 2 )Γ(α+1) (2n+2)Γ(α+n+ 1 2 )Γ(α−n+ 1 2 ) (1.13) a. where, Re(α) > −1; n = 1,2, … … 5. ∫ xλP2γ(x)dx = (−1)γΓΓ 2Γ(γ+ λ 2 + 3 2 )Γ(− λ 2 ) 1 0 , Re(λ) > −1, γ is non-negative integer. (1.14) 6. ∫ xλP2ν+1(x)dx = (−1)ν Γ( 1 2 − λ 2 +ν)Γ(1+ λ 2 ) Γ( 1 2 − λ 2 )Γ(2+ν+ λ 2 ) 1 0 , Re(λ) > −2 (1.15). Integral associated with generalized Laguerre polynomial as: 7. ∫ xβ−1e−xLn (α)(x)dx = Γ(α−β+n+1)Γ(β) Γ(α−β+1) n! , Re(β) > 0, ∞ 0 n is non negative integer. (1.16) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2715 https://internationalpubls.com 8. ∫ uλ−1(1 + u)−μdu = Γ(λ)Γ(μ−1) Γ(μ) ∞ 0 , 0 < Re(λ) < Re(μ). (1.17) 9. ∫ sinμϑ. cosνϑdϑ = π 2 0 Γ( μ+1 2 )Γ( ν+1 2 ) 2Γ( μ+ν+2 2 ) , Re(μ) > 0, Re(ν) > 0. (1.18) 2. Composition of Euler type Integrals: Theorem 1. If Re(η) > −1, Re(ξ) > 0, Re(ν) > 0, Re ( μ ν ) > Re(μ), σ, ρ ∈ C, ξ = ϑ4, Re(σ) > 0, we get following results ∫ e−μt[sinh (νt)]η Mq p p σ∞ 0 (k1, … . , kp, l1 … . lq: ze−ξt)dt = ν−12−η−1 ∏ Γ(lq) q i=1 Γ(1+η) ∏ Γ(kp) p j=1 p+2Ψq+2 [ (k1, 1), … … , (kp, 1), ( μ 2ν − η 2 + ξ 2ν ) , (1,1) (l1, 1), … … ., (lq, 1), (ρ, σ), ( μ 2ν + η 2 + 1 + ξ 2ν ) Z]. (2.1) Proof. To prove above Theorem 1, expressing Mq p p σ is the L.H.S. of (2.1). Changing the order of integral and on evaluating the inner integral with help of (1.10). ∫ e−μt[sinh (νt)]η Mq p p σ∞ 0 (k1, … . , kp, l1 … . lq: ze−ξt)dt = ∫ e−μt[sinh (νt)]η ∑ (k1)m……(kp) m (l1)m…….(lq) m ∞ m=0 zme−ξmt Γ(ρ+σm) dt ∞ 0 = ∑ (k1)m……(kp) m (l1)m…….(lq) m ∞ m=0 zme−ξmt Γ(ρ+σm) ∫ e−(μ+ξm)t[sinh (νt)]ηdt ∞ 0 = ∑ (k1)m……(kp) m (l1)m…….(lq) m ∞ m=0 zme−ξmt Γ(ρ+σm) ν−12−η−1 Γ( μ+ξm 2ν − η 2 )Γ(η+1) Γ( μ+ξm 2ν + η 2 +1) = ν−12−η−1 Γ(η + 1) ∑ (k1)m……(kp) m (l1)m…….(lq) m ∞ m=0 zme−ξmt Γ(ρ+σm) Γ( μ 2ν − η 2 + ξm 2ν ) Γ( μ 2ν + η 2 +1+ ξm 2ν ) . Finally, we use equation (1.3), and we get desired result (2.1). Theorem 2. If Re(ξ) > 0, Re(η) > −1, Re(σ) > 0, Re ( μ 2 ) > Re(η): σ, ρ ∈ C, following results holds: ∫ e−μ[sinh(νt)]η ∞ 0 Mz p(k1, … . , kp, l1 … . lq: z(2 sinh(νt))ξm)p σ dt = ν−12−η−1 ∏ Γ(lq)q j=1 ∏ Γ(kp)p i=1 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2716 https://internationalpubls.com × p+3Ψq+2 [ (k1, 1), … … , (kp, 1), ( μ 2ν − η 2 + ξ 2 ) , (1 + η, ξ), (1,1) (l1, 1), … … ., (lq, 1), (ρ, σ), ( μ 2ν + η 2 + 1 + ξ) ; Z]. (2.2) Proof. For solving the above integral formula (2.2), using definition (1.1) in the L.H.S. of (2.2) and then changing order of integration, we have = ∑ (k1)m … … (kp) m (l1)m … … . (lq) m ∞ m=0 zm2ξm(sinh (νt))ξm Γ(ρ + σm) ∫ e−μ[sinh (νt)]η ∞ 0 dt = ∏ Γ(lq)q j=1 ∏ Γ(kp) p i=1 ∑ Γ(1 + k1) … . . Γ(m + kp) (z2ξ)m Γ(1 + l1) … . . Γ(m + lq)Γ(ρ + σm) ∫ e−μ[sinh (νt)]η+ξm ∞ 0 dt ∞ m=0 = ∏ Γ(lq)q j=1 ∏ Γ(kp) p i=1 ∑ Γ(1 + k1) … . . Γ(m + kp)(z2ξ)m Γ(1 + l1) … . . Γ(m + lq)Γ(ρ + σm) ∞ m=0 ν−12−(η+ξm)−1Γ ( μ 2ν − η + ξm 2 ) Γ(1 + η + ξm) Γ( μ 2ν + η + ξm 2 + 1) = ν−12−η−1 ∏ Γ(lq) q j=1 ∏ Γ(kp) p i=1 ∑ Γ(1+k1)…..Γ(m+kp) Γ(1+l1)…..Γ(m+lq)Γ(ρ+σm) Γ( μ 2ν − η 2 + ξm 2 )Γ(1+η+ξm) Γ( μ 2ν + η 2 +1+ξm) ∞ m=0 zm m! Γ(1 + m). Hence, we call equation (1.3) and then reached at desired result (2.2). Theorem 3. If Re(ξ) > 0, Re(ν) < 1, Re(μ) > 0, σ, ξ ∈ C, Re(σ) > 0, following integral formula holds ∫ uμ(1 − u2)− ν 2Pη ν(u) 1 0 Mz p(k1, … . , kp, l1 … . lq: zuξ)dup σ = 2ν−1 ∏ Γ(lq)q j=1 ∏ Γ(kp)p i=1 × p+3Ψq+3 [ (k1, 1), … … , (kp, 1), ( μ 2 + 1 2 , ξ 2 ) , (1 + η 2 , ξ 2 ) , (1,1) (l1, 1), … … ., (lq, 1), (ρ, σ), ( μ 2 − η 2 − ν 2 + 1, ξ 2 )( μ 2 + η 2 − ν 2 + 3 2 , ξ 2 ) ; Z]. (2.3) 𝐏𝐫𝐨𝐨𝐟. To prove Theorem 3, using definition of generalized M-series in the left hand side of (2.3). After simple simplification, we get ∫ uμ(1 − u2)− ν 2Pη ν(u) 1 0 Mz p(k1, … . , kp, l1 … . lq: zuξ)dup σ Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2717 https://internationalpubls.com = ∑ (k1)m … … (kp) m (l1)m … … . (lq) m ∞ m=0 zm Γ(ρ + σm) ∫ uμ+ξm(1 − u2)− ν 2Pη ν(u)du 1 0 = ∏ Γ(lq)q j=1 ∏ Γ(kp)p i=1 ∑ Γ(1 + k1) … . . Γ(m + kp) Γ(1 + l1) … . . Γ(m + lq)Γ(ρ + σm) 2ν−1Γ ( 1 2 + μ + ξm 2 ) Γ (1 + μ + ξm 2 ) Γ(1 + m)zm Γ (1 + μ + ξm 2 − η 2 − ν 2 ) Γ( μ + ξm 2 + η 2 − ν 2 + 3 2)m! ∞ m=0 Hence Proved. Theorem 4. If Re(ξ) > 0, Re(ν) < 1, Re(μ + ν + η) > 0, Re(μ + ν − η) > 1, Re(σ) > 0, following integral formulas holds: ∫ u−μ(u2 − 1)− ν 2Pη ν(u) ∞ 0 Mz p (k1, … . , kp, l1 … . lq: zu−ξ)dup σ = ∏ Γ(lq) q j=1 (π) 1 2 ∏ Γ(kp) p i=1 p+3Ψq+2 [ (k1, 1), … … , (kp, 1), ( μ+ν+2 2 , ξ 2 ) , ( μ+ν−η−1 2 , ξ 2 ) , (1,1) (l1, 1), … … ., (lq, 1), (ρ, σ), (μ, ξ) ; Z]. (2.4) Proof. Using equation (1.2) and changing the order of integration in the left hand side of (2.4), we have = ∏ Γ(lq)q j=1 ∏ Γ(kp)p i=1 ∑ Γ(1 + k1) … . . Γ(m + kp) zm Γ(1 + l1) … . . Γ(m + lq)Γ(ρ + σm) ∞ m=0 ∫ u−(μ+ξm)(u2 − 1)− ν 2Pη ν(u)du ∞ 0 = ∏ Γ(lq)q j=1 (μ) 1 2 ∏ Γ(kp)p i=1 ∑ Γ(1 + k1) … . . Γ(m + kp) zm Γ(1 + l1) … . . Γ(m + lq)Γ(ρ + σm) ∞ m=0 × Γ( μ+ξm+ν+2 2 )Γ( μ+ξm+ν−η−1 2 )Γ(1+m) Γ(μ+ξm)m! Thus, from equation (1.3), reached at required result of Theorem 4. Theorem 5. If Re(μ) > −1, Re(λ) > 0, Re(σ) > 0, n = 1,2. ,, following integral formula holds: ∫ (1 − u) 1 2(1 + u)μUn(u) 1 0 Mz p(k1, … . , kp, l1 … . lq: z(1 + x)λ)dup σ = (π) 1 22 2n+ 3 2{(n+1)!}2 (2n+2) ∏ Γ(lq) q j=1 ∏ Γ(kp) p i=1 p+3Ψq+2 [ (k1, 1), … … , (kp, 1), ( 2μ+1 2 , λ) , (μ, λ), (1,1) (l1, 1), … … ., (lq, 1), (ρ, σ), (μ + 5 2 , λ) , (μ` − n + 1 2 , λ) ; Z] . (2.5) Proof. To prove above Theorem, using definition (1.1) and relation (1.3) in the left hand side of Theorem 5, and then Simply, we get required result. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2718 https://internationalpubls.com = ∏ Γ(lq) q j=1 ∏ Γ(kp) p i=1 ∑ Γ(1+k1)…..Γ(m+kp) ( z 2λ )m Γ(1+l1)…..Γ(m+lq)Γ(ρ+σm) ∞ m=0 ∫ (1 − u) 1 2(1 + u)μ+λnUn(u) 1 0 du = ∏ Γ(lq) q j=1 ∏ Γ(kp) p i=1 ∑ Γ(1+k1)…..Γ(m+kp) ( z 2λ )m Γ(1+l1)…..Γ(m+lq)Γ(ρ+σm) ∞ m=0 (π) 1 22 μ+λn+2n+ 3 2{(n+1)!}2 (2n+2)Γ(μ+λm+ 5 2 )Γ(μ+λm− n 2 + 1 2 ) = (π) 1 222n+ 3 2{(n + 1)!}2 (2n + 2) ∏ Γ(lq)q j=1 ∏ Γ(kp)p i=1 ∑ Γ(1 + k1) … . . Γ(m + kp) ( z γ ) m Γ(1 + l1) … . . Γ(m + lq)Γ(ρ + σm) ∞ m=0 × Γ (μ + 1 2 + λm) Γ(μ + λm)zm Γ (μ + 5 2 + λm) Γ(μ` − n + 1 2 + λm) Theorem 6. LetRe(μ) > 0, Re(σ) > 0, Re(α − μ + n) > −1, Re(α − μ) > −1. The identity holds: ∫ uμ−1e−uLn α (u) Mz p(k1, … . , kp, l1 … . lq: z(2u)ξ)dup σ∞ 0 = ∏ Γ(lq) q j=1 ∏ Γ(kp) p i=1 p+3Ψq+2 [ (k1, 1), … … , (kp, 1), (α − μ + n + 1, −ξ), (μ, ξ), (1,1) (l1, 1), … … ., (lq, 1), (ρ, σ), (α − μ + 1, −ξ) ; Z2ξ]. (2.6) Proof. Using (1.1) in the left hand side of (2.6) and simple simplification, we get following steps ∫ uμ−1e−uLn α (u) Mz p (k1, … . , kp, l1 … . lq: z(2u)ξ)dup σ ∞ 0 = ∑ (k1)1 … . . (kp)m zm2ξm (l1)1 … . . (lq)mΓ(ρ + σm) ∞ m=0 ∫ uμ+ξme−uLn α (u)du ∞ 0 = ∏ Γ(lq) q j=1 ∏ Γ(kp) p i=1 ∑ Γ(1+k1)…..Γ(m+kp) (z2ξ)m Γ(1+l1)…..Γ(m+lq)Γ(ρ+σm) ∞ m=0 Γ(α−(μ+ξm)+n+1)Γ(μ+ξm)Γ(1+m) Γ((α−μ−ξm+1))n! Hence proved. Theorem 7. If Re(σ) > 0, 0 < Re(λ) < Re(μ), following compostion formula holds ∫ uλ−1(1 + u)−μ Mz p(k1, … . , kp, l1 … . lq: z(1 + u)−ν)dup σ∞ 0 = ∏ Γ(lq) q j=1 ∏ Γ(kp) p i=1 p+2Ψq+2 [ (k1, 1), … … , (kp, 1), (μ − λ, ν), (1,1) (l1, 1), … … ., (lq, 1), (ρ, σ), (μ, ν) ; Z]. (2.7) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2719 https://internationalpubls.com Proof. For evaluating (2.7), applying definition of generalized M-series in the left hand side of (2.7), we get = ∑ (k1)1 … . . (kp)m zm (l1)1 … . . (lq)mΓ(ρ + σm) ∞ m=0 ∫ uλ−1(1 + u)−μ−2mdu ∞ 0 = ∑ (k1)1 … . . (kp)m zm (l1)1 … . . (lq)mΓ(ρ + σm) ∞ m=0 Γ(λ)Γ(μ + νm − λ) Γ(μ + νm) = ∏ Γ(lq)Γ(λ)q j=1 ∏ Γ(kp) p i=1 ∑ Γ(1+k1)…..Γ(m+kp) zm Γ(1+l1)…..Γ(m+lq)Γ(ρ+σm) Γ(μ+νm−λ)Γ(1+m) Γ(μ+νm)m! ∞ m=0 Hence Proved. Theorem 8. Let Re(μ) > 0, Re(ν) > 0, Re(σ) > 0, Re(η) > 0, Re(ξ) > 0. Then ∫ sinμϑ. cosνϑ Mz p(k1, … . , kp, l1 … . lq: z{sinηϑ. cosξϑ})dup σ dϑ π 2 0 = ∏ Γ(lq) q j=1 2 ∏ Γ(kp) p i=1 p+3Ψq+2 [ (k1, 1), … … , (kp, 1), ( μ+1 2 , η 2 ) ( ν+1 2 , ξ 2 ), (1,1) (l1, 1), … … ., (lq, 1), (ρ, σ), ( μ+ν 2 , η+ξ 2 ) ; Z]. (2.8) Proof. Proof of Theorem 8 is same as the roof of Theorem 7. Theorem 9. If Re(ξ) > 0, Re(ν) < 1, Re(μ) > 0, σ, ξ ∈ C, Re(σ) > 0, following integral formula holds ∫ uμ−1e− au 2 Wη,v Mq σ(k1, … . , kp, l1 … . lq: zuξ)dup ρ ∞ 0 = ∏ Γ(lq) q j=1 ∏ Γ(kp) p i=1 aμ p+3Ψq+2 [ (k1, 1), … … , (kp, 1), (μ + ν + 1 2 , ξ) (μ − ν + 1 2 , ξ) , (1,1) (l1, 1), … … ., (lq, 1), (ρ, σ), (μ − k + 1, ξ) ; Z aξ ] (2.9) Proof. For evaluating (2.7), applying definition of generalized M-series in the left hand side of (2.7), we get = ∏ Γ(lq)q j=1 ∏ Γ(kp)p i=1 aμ ∑ Γ(1 + k1) … . . Γ(m + kp) zm Γ(1 + l1) … . . Γ(m + lq)Γ(ρ + σm) ∞ m=0 ∫ uμ+ξe− au 2 Wη,vdu ∞ 0 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2720 https://internationalpubls.com = ∏ Γ(lq) q j=1 ∏ Γ(kp)p i=1 aμ ∑ Γ(1 + k1) … . . Γ(m + kp) ( z aξ)m Γ(1 + l1) … . . Γ(m + lq)Γ(ρ + σm) ∞ m=0 Γ (μ + ξm + ν + 1 2) Γ(μ + ξm − v + 1 2) Γ(1 − k + μ + ξm) Hence proved. 3. Special cases: Corollary 1. If we take p = q = 0 in Theorem 1 and using formula (1.6) , we get the following result ∫ e−μt[sinh(νt)]η∞ 0 Eρ,σ(ze−ξt)dt = ν−12−η−1 Γ(1+η) 1 2Ψ2 [ ( μ 2ν − η 2 , ξ 2ν ) , (1,1) (ρ, σ) , ( μ 2ν + η 2 + 1 + ξ 2ν ) Z]. Corollary 2. If we take If p = 0, q = 1 in Theorem 1 and using (1.7), we get following result. ∫ e−μt[sinh(νt)]η∞ 0 Wρ,σ(ze−ξt)dt = ν−12−η−1 Γ(1+η) 1 1Ψ2 [ ( μ 2ν − η 2 , ξ 2ν ) , (ρ, σ) , ( μ 2ν + η 2 + 1 + ξ 2ν ) Z]. Corollary 3. If we take If p = q = 1, k = a and l = 1, an equation (2.3) and using (1.8), we get ∫ uμ(1 − u2)− ν 2Pη ν(u) 1 0 Eρ,σ a (zuξ)du = 2ν−1 ∏ Γ(lq) q j=1 ∏ Γ(kp) p i=1 3Ψ3 [ (a, 1) , ( μ 2 + 1 2 , ξ 2 ) , ( μ+1 2 , ξ 2 ) (ρ, σ) , ( μ 2 − η 2 − ν 2 + 1, ξ 2 )( μ 2 + η 2 − ν 2 + 3 2 , ξ 2 ) ; Z]. Corollary 4. If we take ρ = σ = 1 in Theorem 3 and using (1.5), we get ∫ uμ(1 − u2)− ν 2Pη ν(u) 1 0 pFq [ k1, … . . kp; l1, … … . lq; z] du = 2ν−1 ∏ Γ(lq) q j=1 ∏ Γ(kp) p i=1 p+2Ψq+2 [ (k1, 1), … … , (kp, 1), ( μ 2 + 1 2 , ξ 2 ) , (1 + μ 2 , ξ 2 ) (l1, 1), … … ., (lq, 1), ( μ 2 − η 2 − ν 2 + 1, ξ 2 )( μ 2 + η 2 − ν 2 + 3 2 , ξ 2 ) ; Z]’ Corollary 5. If we take p = q = 0 in Theorem 4 and using (1.6) , we get following result ∫ u−μ(u2 − 1)− ν 2Pη ν(u) ∞ 0 Eρ,σ(zu−ξ)du = 1 (π) 1 2 3Ψ2 [ ( μ+ν+2 2 , ξ 2 ) , ( μ+ν−η−1 2 , ξ 2 ) , (1,1) , (ρ, σ), (μ, ξ) ; Z]. Corollary 6. If we take p = 0, q = 1 in Theorem 5 and using (1.8), we get Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2721 https://internationalpubls.com ∫ (1 − u) 1 2(1 + u)μUn(u) 1 0 Wρ,σ (z ( 1+μ 2 ) λ ) du = (π) 1 22 2n+ 3 2{(n+1)!}2 (2n+2) 2Ψ3 [ ( 2μ+1 2 , λ) , (μ, λ) (ρ, σ) , (μ + 5 2 , λ) , (μ` − n + 1 2 , λ) ; Z] . Corollary 7. If we take p = q = 1, k=1 in Theorem 8 and (1.7), we get ∫ sinμϑ. cosνϑEρ,σ a (zsinμϑ. cosνϑ)dϑ = Γ(L) 2Γ(K) 3Ψ2 [ (a, 1), ( μ+1 2 , η 2 ) ( ν+1 2 , ξ 2 ) (ρ, σ) , ( μ+ν 2 , η+ξ 2 ) ; Z] π 2 0 . Corollary 8. If we take p = q = 0. In Theorem 9 and using (1.6) , we get ∫ 𝑢𝜇−1𝑒− 𝑎𝑢 2 𝑊𝜂,𝑣(𝑎𝑢)𝐸𝜌,𝜎(𝑧𝑢𝜉) ∞ 0 𝑑𝑢 = 𝑎𝜇 3𝛹2 [ (𝜇 + 𝜈 + 1 2 , 𝜉) , (𝜇 − 𝜈 + 1 2 , ξ), (1,1) (𝜌, 𝜎), (𝜇 − 𝑘 + 1, 𝜉) ; Z/aξ] 4. Conclusion remark In this investigation, we have managed to obtain new forms for the Eulerian type integrals that are associated with the generalized M-series, which in turn has enriched the theory of the special functions. The integral expressions derived not only extend our knowledge of the structural properties of the generalised M-series, but also reveal the flexibility by analysing special cases involving special parameter values. These results offer opportunities for possible applications in a broad range of disciplines such as mathematical physics, engineering, and applied mathematics. Future research may explore further generalizations and applications of these integrals in solving complex problems across different scientific disciplines. We concluded from the present research work by giving some comments on the results of Theorem 1-Theorem 9and their corollaries. The integrals can be further generalized and applied in solving some long-standing problems in that has been presented in this letter. We ended present research work with comments of Theorem 1- Theorem 9and their corollaries. The proposed generalized M-series is an interesting function which is equivalent to one of the several families of transcendental and special functions namely exponential function, binomial series, cosine function, sine function, Mittag-leffer function, Wright function, gauss hypergeometric function, Fox-H function and Meijer G-function which appears to be new even in the case of special cases. Thus, we can deduce more useful results and their equivalent forms from Theorem 1 to Theorem 9 in terms of Fox H-function and Meijer G-function. Acknowledgement: All authors would like to thanks integral University, Lucknow, India for providing the manuscript (MCN): IU/R&D/2025-MCN0003703 for this work. Conflict of interest: The authors declare that there is no conflict of interest. References [1] A. Erdelyi, Transformation of hypergeometric function of two variables, Proc. Roy. Soc. Edinburg A, 62 , 378-385, 1948. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2722 https://internationalpubls.com [2] M. Kamarujjama, O. Khan, Computation of new class of integrals involving generalized Galue type Struve function, J. Comput. Appl. Math., 351 (2019) 228-236. [3] N. U. Khan, M. Iqbal Khan, O. Khan, Certain finite integrals involving generalized Wright function, Advanced Mathematical Models & Applications, 6, (3), (2021), 292- 301 [4] N. Khan, S. Husain, O. Khan, A novel kind of beta logarithmic function and their properties, Hacet. J. Math. Stat. 52 (4) (2023), 945–955 [5] O. Khan, M. Kamarujjama, N.U. Khan, D. Baleanu, K.S. Nisar, Computable solution of fractional kinetic equations using Mathieu-type series, Adv. Differ Equ. 2019, 234 (2019). [6] E. Ilhan and I.O. Kiymaz, A generalization of truncated M-fractional derivatives and applications to fractional differential equation, Appl. Math. Nonlinear Sci., 5 (1) (2020), pp. 171-188. [7] E.D.,Rainville,:Special Functions, The Macmillan Company, New York, 2013. [8] M. Kamarujjama, N. U. Khan, O. Khan, Juan J. Nieto, Extended Type k-Mittag–Leffler Function and Its Applications, Int. J. Appl. Comput. Math (2019): 72, 1-14. https://doi.org/10.1007/s40819-019-0656-5. [9] S. Kumar, O. Khan, N.U. Khan, Caputo Derivative Formulas of Hurwitz-Lerch Zeta Function and Applications, Communication on Applied Nonlinear analysis., 32 (10) (2025), 2434-2441. [10] D.S. Sachan, D. Kumar, K.S., Nisar, Certain Properties Associated with Generalized M- Series using Hadamard Product, Sahand Communications in Mathematical Analysis, 21 (1) (2024), pp. 151-171. [11] D.S. Sachan, H. Jalori and S. Jaloree, Fractional calculus of product of M-series and I- function of two variables, Jnanabha, 52 (1) (2022), pp. 189-202. [12] M. Sharma, R. Jain, A note on generalized M-series, Fract. Calc. Appl. Anal. 12 (1), 2009, 449-452. [13] A.K. Shukla, A.K., and J.C. Prajapati. On a generalization of Mittag-Leffler function and its prop-erties. Journal of Mathematical Analysis and Applications 336 (2) (2007): 797–811. [14] D.L. Suthar, H. Tadesse and K. Tilahun, Integrals involving Jacobi polynomials and M- Series, J. Fract. Calc. Appl., 9 (2) (2018), 287-294. . [15] I.N. Sneddon, The use of integral transforms. New York: Tata McGraw-Hill, 1979. [16] M.R. Spiegel, Theory and problem of Laplace transforms, Schums Outline Series. New york: McGraw-Hill, 1965. [17] H.M. Srivastava, and P.W. Karlsson, Multiple Gaussian Hypergeometric Series, HalstedPress(Ellis Horwood Limited, Chichester). New York: Wiley, 1985. [18] A. Wiman, Uber den fundamental satz in der theorie der funktionen Acta Mathematica29 (1): 191–201, 1905. https://doi.org/10.1007/s40819-019-0656-5 https://www.scopus.com/record/display.uri?eid=2-s2.0-85182790769&origin=reflist&sort=plf-f&src=s&imp=t&sid=2d344b666f7fe6dedcff0518b8ad3e75&sot=cite&sdt=a&sl=23&s=REF%282-s2.0-85182790769%29 https://www.scopus.com/record/display.uri?eid=2-s2.0-85182790769&origin=reflist&sort=plf-f&src=s&imp=t&sid=2d344b666f7fe6dedcff0518b8ad3e75&sot=cite&sdt=a&sl=23&s=REF%282-s2.0-85182790769%29