Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2434 https://internationalpubls.com Caputo Derivative Formulas of Hurwitz-Lerch Zeta Function and Applications Sandeep Kumar1, Owais Khan2*, N. U. Khan3 and N. Ahmad4 1,2,4Department of Mathematics and Statistics, Integral University, Lucknow-226026,India Emails: 1sandeep8603@gmail.com, 2owkhan05@gmail.com, 4najmuddinahmad33@gmail.com 3Department of Applied Mathematics, Aligarh Muslim University, Aligarh-202002, India Email:3nukhanmath@gmail.com Article History: Received: 12-01-2025 Revised: 15-02-2025 Accepted: 01-03-2025 Abstract: In this paper, we find the fractional derivative formulas of Hurwitz-Lerch Zeta function. Further, we compute the solution of fractional differential equations involving Hurwitz-Lerch Zeta function. Keywords: Hurwitz-Lerch Zeta function, hypergeometric function and Fractional derivatives. 1. Introduction Fractional Calculus serves as an excellent tool for studying fractional order integrals and derivatives. There are a lot of disciplines in science and engineering that benefit from fractional calculus. In a variety of fields, fractional differential equations and their applications have played a significant role. These include applied science, physics, biology, chemistry and engineering science. As a system of differential, kinetic equations provide a description of the rate at which changes in the chemical composition of a star occur. Fractional differential equations have been widely and successfully used to describe and solve many problems in physics and astrophysics over the past several decades. In mathematics and mathematical physics, the special functions are useful for the solution of fractional integral and differential equation problems. In order to incorporate fractional derivatives into differential equations, fractional differential equations were developed. When f(w) is a function of order alpha. Its fractional derivatives are represented mathematically as Dαf(w), where α is a non integer. Many fractional derivatives exist, including Riemann-Lionville, Caputo and Grunwald-Letnikov derivatives. They each have their own advantage and uses. The fractional derivative of f(w) in the Caputo sense is defined as: Dαf(w) = Il−αDlf(w) (1) Dαf(ω) = 1 Γ(l−α) ∫ (ω − u)l−α−1f lw 0 (u)du. . (2) For l − 1 ˂ α ≤ l, l ∈ N, ω ˃ 0 . The Caputo derivative, we have DαC = 0, C is constant. mailto:1sandeep8603@gmail.com mailto:2owkhan05@gmail.com mailto:4najmuddinahmad33@gmail.com mailto:3nukhanmath@gmail.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2435 https://internationalpubls.com Dαrr = { 0 , r ≤ α − 1 Γ(r+1) Γ(r−α+1) tr−α , r ˃ (α − 1) . (3) Mathematical special functions or SFs date back to the nineteenth century, when they were developed as a unified and complete theory. Researchers and engineers working with differential equations are well aware of the value of SFs as a tool for mathematical analysis. Several of these named functions are formulated as mathematical models by solving differential equations and systems of integer order. As a result of the growing by interest and widespread application of differential equations and fractional order system, many physical, engineering, automation, biological, chemical, earth science, economic phenomena have been better represented in the function, Euler beta function, and many more function have all recently seen extensions created by numerous writers. We are familiar with the Hurwitz-Lerch zeta function φ(w, k, r) defined as: 𝜑(𝑤, 𝑘, 𝑡) = ∑ 𝑤𝑚 (𝑚+𝑟)𝑘 ,∞ 𝑚=0 (4) (𝑟 ∈ 𝑍+: 𝑘 ∈ 𝐶 𝑤ℎ𝑒𝑛 |𝑤| < 1: 𝑅(𝑘) > 1 𝑤ℎ𝑒𝑛 |𝑤| = 1). Various generalizations of the Hurwitz-Lerch zeta functions have been given by the researchers. For example, Goyal et.al. has introduced an extension of Hurwitz-Lerch zeta function defined as: 𝜑𝜃1 ∗ (𝑤, 𝑘, 𝑡) = ∑ (𝜃1)𝑚 𝑚! 𝑤𝑚 (𝑚 + 𝑟)𝑘 , (5) ∞ 𝑚=0 (𝜃1 ∈ 𝐶; 𝑟 ∈ 𝑍+; 𝑘 ∈ 𝐶 𝑖𝑓 |𝑤| < 1 ; 𝑅(𝑘 − 𝛿) > 1 𝑖𝑓 |𝑤| = 1). Lin et. al. [13] also defined the Hurwitz-Lerch zeta function as: 𝜑𝜃1,𝜃2, 𝛽,𝛿 (𝑤, 𝑘, 𝑝) = ∑ (𝜃1)𝛽𝑚 (𝜃2)𝛿𝑚 𝑤𝑚 (𝑚 + 𝑝)𝑘 , (6) ∞ 𝑚=0 (𝜃1 ∈ 𝐶; 𝑝, 𝜃2 ∈ 𝑍+; 𝛽, 𝑤 ∈ 𝑅 +; 𝛽 < 𝑤 𝑖𝑓𝑠, 𝑤 ∈ 𝐶; 𝛽 = 𝛿, 𝑠 𝜖 𝐶 𝑖𝑓 |𝑤| < 1 ; 𝑅(𝑠 − 𝜃1 + 𝜃2) > 1 , |𝑤| = 1) Garg et. al. [3] also introduced Hurwitz-Lerch zeta function as: 𝜑𝜃1,𝜃2,𝜃3 𝛽,𝛾,𝛿 (𝑤, 𝑠, 𝑝) = ∑ (𝜃1)𝑟𝛽 (𝜃2)𝑟𝛿 (𝜃2)𝑟𝛾(𝑤𝛼)𝑟 (𝑟 + 𝑝)𝑠 , (7) ∞ 𝑚=0 (𝜃1, 𝜃2 ∈ 𝐶; 𝜃3, 𝑝 ∈ 𝑍+; 𝑠 ∈ 𝐶; 𝑖𝑓 |𝑤| < 1 ; 𝑅(𝑠 + 𝜃3 − 𝜃1 − 𝜃2) > 1 , |𝑤| = 1) In order to address the short comings associated with fractional derivatives in mathematics and physics, the author developed formulas for the Caputo derivative of the Hurwitz-Lerch Zeta function. As a result of these derivatives formulas, we are able to compute solutions to fractional differential equations. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2436 https://internationalpubls.com 2. Analysis of Method In mathematics and mathematics physics, the special functions are useful for the solution of fractional integral and differential equation problems. In order to incorporate fractional derivatives into differential equations, fractional differential equations were developed [2, 12, 15]. The Hurwitz- Lerch zeta function defined by power series (7) has efficiency as solution of fractional order differential and integral equations and thus have important role of the fractional calculus theory and applications. In this section, we consider few examples that demonstrate the performance and efficiency of Hurwitz-Lerch zeta function for solving linear fractional differential equations with fractional derivatives. The Hurwitz-Lerch zeta function suggests that the linear term $y(x)$ is decomposed by an power series: 𝑓(𝑤) = 𝜑𝜃1,𝜃2,𝜃3 𝛽,𝛾,𝛿 (𝑤, 𝑠, 𝑝) = ∑ (𝜃1)𝑟𝛽 (𝜃3)𝑟𝛿 (𝜃2)𝑟𝛾 (𝑟+𝑝)𝑠 (𝐴𝑤𝛼)𝑟 (8)∞ 𝑟=0 𝑓(𝑤) = 1 𝑝𝑠 + (𝜃1)𝛽 (𝜃3)𝛿 (𝜃2)𝛾 (1 + 𝑝)𝑠 (𝐴𝑤𝛼)1+ . . . . (9) Theorem 1. The following derivative formula holds: 𝐷𝛼𝑓(𝑤) = ∑ (𝜃1)𝑟𝛽 (𝜃3)𝑟𝛿 (𝜃2)𝑟𝛾 (𝑟+𝑝)𝑠 𝐴𝑟𝛤(𝑟𝛼+1) 𝛤(𝛼(𝑟−1)+1) 𝑤𝛼(𝑟−1)∞ 𝑟=1 (10) Proof. From (1) and (4), we have 𝐷𝛼𝑓(𝑤) = 1 𝛤(𝑙 − 𝛼) ∫(𝑤 − 𝑢)𝑙−𝛼−1 𝑤 0 ∑ (𝜃1)𝑟𝛽 (𝜃3)𝑟𝛿 (𝜃2)𝑟𝛾 (𝑟 + 𝑝)𝑠 𝐴𝛼𝐷𝑡𝑢𝑟𝛼𝑑 ∞ 𝑟=1 = 1 𝛤(𝑙 − 𝛼) ∫(𝑤 − 𝑢)𝑙−𝛼−1 𝑤 0 ∑ (𝜃1)𝑟𝛽(𝜃2)𝑟𝛾 (𝜃3)𝑟𝛿 𝐴𝛼 (𝑟 + 𝑝)𝑠 𝛤(𝑟𝛼 + 1) 𝛤(𝑟𝛼 − 𝑙 + 1) 𝑢𝑟𝛼−𝑙𝑑𝑢 ∞ 𝑟=1 = 1 𝛤(𝑙 − 𝛼) ∑ (𝜃1)𝑟𝛽(𝜃2)𝑟𝛾 (𝜃3)𝑟𝛿 𝐴𝛼 (𝑟 + 𝑝)𝑠 𝛤(𝑟𝛼 + 1) 𝛤(𝑟𝛼 − 𝑙 + 1) ∞ 𝑟=1 ∫(𝑤 − 𝑢)𝑙−𝛼−1 𝑤 0 𝑢𝑟𝛼−𝑙𝑑𝑢 = 1 𝛤(𝑙 − 𝛼) ∑ (𝜃1)𝑟𝛽(𝜃2)𝑟𝛾 (𝜃3)𝑟𝛿 𝐴𝛼 (𝑟 + 𝑝)𝑠 𝛤(𝑟𝛼 + 1) 𝛤(𝑟𝛼 − 𝑙 + 1) ∞ 𝑟=1 ∫ 𝑤𝑙−𝛼−1(1 − 𝑢 𝑤 )𝑙−𝛼−1 𝑤 0 𝑢𝑟𝛼−𝑙𝑑𝑢 Now let 𝑢 𝑤 = 𝑣 𝑑𝑢 = 𝑤𝑑𝑣. Then = 1 𝛤(𝑙 − 𝛼) ∑ (𝜃1)𝑟𝛽(𝜃2)𝑟𝛾 (𝜃3)𝑟𝛿 𝐴𝛼 (𝑟 + 𝑝)𝑠 𝛤(𝑟𝛼 + 1) 𝛤(𝑟𝛼 − 𝑙 + 1) ∞ 𝑟=1 𝑤𝛼(𝑟−1) 𝛤(𝑟𝛼 − 𝑙 + 1)𝛤(𝑙 − 𝛼) 𝛤(𝛼(𝑟 − 1) + 1) On solving we get the desired result (10). Theorem 2. The following derivative formula holds: Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2437 https://internationalpubls.com 𝐷2𝛼𝑓(𝑤) = ∑ (𝜃1)𝑟𝛽 (𝜃3)𝑟𝛿 (𝜃2)𝑟𝛾 (𝑟+𝑝)𝑠 𝐴𝑟𝛤(𝑟𝛼+1) 𝛤(𝛼(𝑟−2)+1) 𝑤𝛼(𝑟−2)∞ 𝑟=2 . (11) Similarly, we can proof the above result as a proof of Theorem 1. 3. Numerical applications In this section, we consider few examples that demonstrate the extended Hurwitz-Lerch Zeta function for solving linear differential equation with fractional derivative. Example 1. The solution of following fractional differential equation 𝐷𝛼𝑓(𝑤) − 𝐶𝑓(𝑤) = 0. By equation (4) and theorem (1) ∑ (𝜃1)𝑟𝛽 (𝜃3)𝑟𝛿 (𝜃2)𝑟𝛾 (𝑟 + 𝑝)𝑠 𝐴𝑟𝛤(𝑟𝛼 + 1) 𝛤(𝛼(𝑟 − 1) + 1) 𝑤𝛼(𝑟−1) ∞ 𝑟=1 − 𝐶 ∑ (𝜃1)𝑟𝛽 (𝜃3)𝑟𝛿 (𝜃2)𝑟𝛾 (𝑟 + 𝑝)𝑠 (𝐴𝑤𝛼)𝑟 = 0 ∞ 𝑟=0 Replace r by r+1 in first summation ∑ (𝜃1)(𝑟+1)𝛽 (𝜃3)(𝑟+1)𝛿 (𝜃2)(𝑟+1)𝛾 (𝑟 + 1 + 𝑝)𝑠 𝐴𝑟+1𝛤((𝑟 + 1)𝛼 + 1) 𝛤(𝛼𝑟 + 1) 𝑤𝛼𝑟 ∞ 𝑟=0 − 𝐶 ∑ (𝜃1)𝑟𝛽 (𝜃3)𝑟𝛿 (𝜃2)𝑟𝛾 (𝑟 + 𝑝)𝑠 (𝐴𝑤𝛼)𝑟 = 0 ∞ 𝑟=0 ∑[ (𝜃1)(𝑟+1)𝛽 (𝜃3)(𝑟+1)𝛿 (𝜃2)(𝑟+1)𝛾 (𝑟 + 1 + 𝑝)𝑠 𝐴1𝛤((𝑟 + 1)𝛼 + 1) 𝛤(𝛼𝑟 + 1) 𝑤𝛼𝑟 ∞ 𝑟=0 − 𝐶 ∑ (𝜃1)𝑟𝛽 (𝜃3)𝑟𝛿 (𝜃2)𝑟𝛾 (𝑟 + 𝑝)𝑠 ]𝐴𝑟𝑤𝑟𝛼 = 0 ∞ 𝑟=0 Now equating to zero the coefficient of 𝑤𝑟𝛼 , we get (𝜃1)(𝑟+1)𝛽(𝜃2)(𝑟+1)𝛾 (𝜃3)(𝑟+1)𝛿 𝐴 (𝑟 + 1 + 𝑝)𝑠 𝛤((𝑟 + 1)𝛼 + 1) 𝛤(𝛼𝑟 + 1) = 𝐶 (𝜃1)𝑟𝛽 (𝜃3)𝑟𝛿 (𝜃2)𝑟𝛾 (𝑝 + 𝑟)𝑠 𝑎𝑡 𝑟 = 0, (𝜃1)𝛽(𝜃2)𝛾 (𝜃3)𝛿(𝑝 + 1)𝑠 𝛤(𝛼 + 1) 𝛤(1) 𝐴 = 𝐶 1 𝑝𝑠 (𝜃1)𝛽(𝜃2)𝛾 (𝜃3)𝛿(𝑝 + 1)𝑠 𝐴 = 𝐶 1 𝑝𝑠𝛤(𝛼 + 1) 𝑎𝑡 𝑟 = 1, (𝜃1)2𝛽(𝜃2)2𝛾 (𝜃3)2𝛿(𝑝 + 2)𝑠 𝛤(2𝛼 + 1) 𝛤(𝛼 + 1) 𝐴 = 𝐶 (𝜃1)𝛽(𝜃2)𝛾 (𝜃3)𝛿(𝑝 + 1)𝑠 (𝜃1)2𝛽(𝜃2)2𝛾 (𝜃3)2𝛿(𝑝 + 2)𝑠 𝛤(2𝛼 + 1) 𝛤(𝛼 + 1) 𝐴𝐴 = 𝐶 (𝜃1)𝛽(𝜃2)𝛾 (𝜃3)𝛿(𝑝 + 1)𝑠 𝐴 (𝜃1)2𝛽(𝜃2)2𝛾 (𝜃3)2𝛿(𝑝 + 2)𝑠 𝛤(2𝛼 + 1) 𝛤(𝛼 + 1) 𝐴𝐴 = 𝐶𝐶 1 𝑝𝑠𝛤(𝛼 + 1) = 𝐶2 1 𝑝𝑠𝛤(𝛼 + 1) 𝑡ℎ𝑢𝑠 (𝜃1)2𝛽(𝜃2)2𝛾 (𝜃3)2𝛿(𝑝 + 2)𝑠 𝐴2 = 𝐶2 1 𝑝𝑠𝛤(2𝛼 + 1) 𝑎𝑡 𝑟 = 2, (𝜃1)3𝛽(𝜃2)3𝛾 (𝜃3)3𝛿(𝑝 + 3)𝑠 𝛤(3𝛼 + 1) 2𝛤(𝛼 + 1) 𝐴 = 𝐶 (𝜃1)2𝛽(𝜃2)2𝛾 (𝜃3)2𝛿(𝑝 + 2)𝑠 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2438 https://internationalpubls.com 𝑜𝑛 𝑠𝑜𝑙𝑣𝑖𝑛𝑔 (𝜃1)3𝛽(𝜃2)3𝛾(𝜃3)3𝛿(𝑝 + 3)𝑠 𝐴 = 𝐶3 1 𝑝𝑠𝛤(3𝛼 + 1) Substituting these values in equation (5), we get 𝑓(𝑤) = 1 𝑝𝑠 + 𝐶 1 𝑝𝑠𝛤(𝛼 + 1) 𝑤𝛼 + 𝐶2 1 𝑝𝑠𝛤(2𝛼 + 1) 𝑤2𝛼 + ⋯ (10) Example 2. Again we take a fractional differential equation 𝐷2𝛼𝑓(𝑤) − 𝐵𝑓(𝑤) = 0 (11) From eq. (4) and theorem (2), we get ∑ (𝜃1)𝑟𝛽 (𝜃3)𝑟𝛿 (𝜃2)𝑟𝛾 (𝑟 + 𝑝)𝑠 𝐴𝑟𝛤(𝑟𝛼 + 1) 𝛤(𝛼(𝑟 − 2) + 1) 𝑤𝛼(𝑟−2) ∞ 𝑟=2 − 𝐵 ∑ (𝜃1)𝑟𝛽 (𝜃3)𝑟𝛿 (𝜃2)𝑟𝛾 (𝑟 + 𝑝)𝑠 𝐴𝑟(𝑤𝛼)𝑟 = 0 ∞ 𝑟=0 Replace r by r+2 in the above equation (only in first summation) then ∑ (𝜃1)(𝑟+2)𝛽 (𝜃3)(𝑟+2)𝛿 (𝜃2)(𝑟+2)𝛾 (𝑟 + 2 + 𝑝)𝑠 𝐴𝑟+2𝛤((𝑟 + 2)𝛼 + 1) 𝛤(𝛼𝑟 + 1) 𝑤𝛼𝑟 ∞ 𝑟=0 − 𝐵 ∑ (𝜃1)𝑟𝛽 (𝜃3)𝑟𝛿 (𝜃2)𝑟𝛾 (𝑟 + 𝑝)𝑠 𝐴𝑟(𝑤𝛼)𝑟 = 0 ∞ 𝑟=0 ∑[ (𝜃1)(𝑟+2)𝛽 (𝜃3)(𝑟+2)𝛿 (𝜃2)(𝑟+2)𝛾 (𝑟 + 2 + 𝑝)𝑠 𝐴2𝛤((𝑟 + 2)𝛼 + 1) 𝛤(𝛼𝑟 + 1) 𝐴2 ∞ 𝑟=0 − 𝐵 (𝜃1)𝑟𝛽 (𝜃3)𝑟𝛿 (𝜃2)𝑟𝛾 (𝑟 + 𝑝)𝑠 ]𝐴𝑟(𝑤𝛼)𝑟 = 0 Equating to zero the coefficient of 𝑤𝑟𝛼 in the above equation (𝜃1)(𝑟+2)𝛽 (𝜃3)(𝑟+2)𝛿 (𝜃2)(𝑟+2)𝛾 (𝑟 + 2 + 𝑝)𝑠 𝛤((𝑟 + 2)𝛼 + 1) 𝛤(𝛼𝑟 + 1) 𝐴2 − 𝐵 (𝜃1)𝑟𝛽 (𝜃3)𝑟𝛿 (𝜃2)𝑟𝛾 (𝑟 + 𝑝)𝑠 = 0 (𝜃1)(𝑟+2)𝛽 (𝜃3)(𝑟+2)𝛿 (𝜃2)(𝑟+2)𝛾 (𝑟 + 2 + 𝑝)𝑠 𝛤((𝑟 + 2)𝛼 + 1) 𝛤(𝛼𝑟 + 1) 𝐴2 = 𝐵 (𝜃1)𝑟𝛽 (𝜃3)𝑟𝛿 (𝜃2)𝑟𝛾 (𝑟 + 𝑝)𝑠 𝑎𝑡 𝑟 = 0, (𝜃1)2𝛽 (𝜃3)2𝛿 (𝜃2)2𝛾 (2 + 𝑝)𝑠 𝛤(2𝛼 + 1) 𝛤(1) 𝐴2 = 𝐵 𝑝𝑠 (𝜃1)2𝛽 (𝜃3)2𝛿 (𝜃2)2𝛾 (2 + 𝑝)𝑠 𝐴2 = 𝐵 𝑝𝑠𝛤(2𝛼 + 1) 𝑎𝑡 𝑟 = 1, (𝜃1)3𝛽 (𝜃3)3𝛿 (𝜃2)3𝛾 (3 + 𝑝)𝑠 𝐴3 = 𝐴𝐵 (𝜃1)𝛽(𝜃2)𝛾 (𝜃3)𝛿(𝑝 + 1)𝑠 𝑎𝑡 𝑟 = 2, (𝜃1)4𝛽 (𝜃3)4𝛿 (𝜃2)4𝛾 (4 + 𝑝)𝑠 𝛤(4𝛼 + 1) 𝛤(2𝛼) 𝐴2 = 𝐵 (𝜃1)2𝛽(𝜃2)2𝛾 (𝜃3)2𝛿(𝑝 + 2)𝑠 Multiplying above equation by A2 (𝜃1)4𝛽 (𝜃3)4𝛿 (𝜃2)4𝛾 (4 + 𝑝)𝑠 𝛤(4𝛼 + 1) 𝛤(2𝛼) 𝐴4 = 𝐵 (𝜃1)2𝛽(𝜃2)2𝛾 (𝜃3)2𝛿(𝑝 + 2)𝑠 𝐴2 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2439 https://internationalpubls.com (𝜃1)4𝛽 (𝜃3)4𝛿 (𝜃2)4𝛾 (4 + 𝑝)𝑠 𝛤(4𝛼 + 1) 𝛤(2𝛼) 𝐴4 = 𝐵2 1 𝑝𝑠𝛤(4𝛼 + 1) Now put these values in equation (5) then, 𝑓(𝑤) = 1 𝑝𝑠 + (𝜃1)𝛽(𝜃2)𝛾 (𝜃3)𝛿(𝑝 + 1)𝑠 𝐴𝑤𝛼 + 𝐵 1 𝑝𝑠𝛤(2𝛼 + 1) 𝑤2𝛼 + 𝐵 (𝜃1)𝛽(𝜃2)𝛾 (𝜃3)𝛿(𝑝 + 1)𝑠 𝐴𝑤3𝛼 + 𝐵2 1 𝑝𝑠𝛤(4𝛼 + 1) 𝑤4𝛼 + ⋯ (12) Example 3. Consider fractional differential equation D2αf(w) + Dαf(w) − 3f(w) = 0 . (13) Then by equation (4), theorem (1) and theorem (2), ∑ (𝜃1)𝑘𝛽 (𝜃3)𝑘𝛿 (𝜃2)𝑘𝛾 (𝑘 + 𝑝)𝑠 𝐴𝑘𝛤(𝑘𝛼 + 1) 𝛤(𝛼(𝑘 − 2) + 1) 𝑤𝛼(𝑘−2) ∞ 𝑘=2 + ∑ (𝜃1)𝑘𝛽 (𝜃3)𝑘𝛿 (𝜃2)𝑘𝛾 (𝑘 + 𝑝)𝑠 𝐴𝑘𝛤(𝑘𝛼 + 1) 𝛤(𝛼(𝑘 − 1) + 1) 𝑤𝛼(𝑘−1) ∞ 𝑘=1 − 3 ∑ (𝜃1)𝑘𝛽 (𝜃3)𝑘𝛿 (𝜃2)𝑘𝛾 (𝑘 + 𝑝)𝑠 𝐴𝑘(𝑤𝛼)𝑘 = 0 ∞ 𝑘=0 Replacing k by k+2 in the first summation and k by k+1 in second summation respectively, ∑ (𝜃1)(𝑘+2)𝛽 (𝜃3)(𝑘+2)𝛿 (𝜃2)(𝑘+2)𝛾 ((𝑘 + 2) + 𝑝)𝑠 𝐴𝑘+2𝛤((𝑘 + 2)𝛼 + 1) 𝛤(𝛼(𝑘) + 1) 𝑤𝛼𝑘 ∞ 𝑘=0 + ∑ (𝜃1)(𝑘+1)𝛽 (𝜃3)(𝑘+1)𝛿 (𝜃2)(𝑘+1)𝛾 (𝑘 + 1 + 𝑝)𝑠 𝐴𝑘+1𝛤((𝑘 + 1)𝛼 + 1) 𝛤(𝛼𝑘 + 1) 𝑤𝛼𝑘 ∞ 𝑘=0 − 3 ∑ (𝜃1)𝑘𝛽 (𝜃3)𝑘𝛿 (𝜃2)𝑘𝛾 (𝑘 + 𝑝)𝑠 𝐴𝑘(𝑤𝛼)𝑘 = 0 ∞ 𝑘=0 ∑[ (𝜃1)(𝑘+2)𝛽 (𝜃3)(𝑘+2)𝛿 (𝜃2)(𝑘+2)𝛾 ((𝑘 + 2) + 𝑝)𝑠 𝐴2𝛤((𝑘 + 2)𝛼 + 1) 𝛤(𝛼(𝑘) + 1) ∞ 𝑘=0 + (𝜃1)(𝑘+1)𝛽 (𝜃3)(𝑘+1)𝛿 (𝜃2)(𝑘+1)𝛾 (𝑘 + 1 + 𝑝)𝑠 𝐴1𝛤((𝑘 + 1)𝛼 + 1) 𝛤(𝛼𝑘 + 1) − 3 (𝜃1)𝑘𝛽 (𝜃3)𝑘𝛿 (𝜃2)𝑘𝛾 (𝑘 + 𝑝)𝑠 ]𝐴𝑘(𝑤𝛼)𝑘 = 0 Now equating to zero the coefficient of 𝑤𝑘𝛼 (𝜃1)(𝑘+2)𝛽 (𝜃3)(𝑘+2)𝛿 (𝜃2)(𝑘+2)𝛾 ((𝑘 + 2) + 𝑝)𝑠 𝐴2𝛤((𝑘 + 2)𝛼 + 1) 𝛤(𝛼(𝑘) + 1) + (𝜃1)(𝑘+1)𝛽 (𝜃3)(𝑘+1)𝛿 (𝜃2)(𝑘+1)𝛾 (𝑘 + 1 + 𝑝)𝑠 𝐴1𝛤((𝑘 + 1)𝛼 + 1) 𝛤(𝛼𝑘 + 1) − 3 (𝜃1)𝑘𝛽 (𝜃3)𝑘𝛿 (𝜃2)𝑘𝛾 (𝑘 + 𝑝)𝑠 = 0 𝑎𝑡 𝑘 = 0, (𝜃1)2𝛽 (𝜃3)2𝛿 (𝜃2)2𝛾 (2 + 𝑝)𝑠 𝐴2𝛤(2𝛼 + 1) 𝛤(1) + (𝜃1)𝛽 (𝜃3)𝛿 (𝜃2)𝛾 (1 + 𝑝)𝑠 𝐴1𝛤(𝛼 + 1) 𝛤(1) − 3 1 𝑝𝑠 = 0 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2440 https://internationalpubls.com 𝑎𝑡 𝑘 = 1, (𝜃1)3𝛽 (𝜃3)3𝛿 (𝜃2)3𝛾 (3 + 𝑝)𝑠 𝛤(3𝛼 + 1) 𝛤(𝛼 + 1) 𝐴3 + (𝜃1)2𝛽 (𝜃3)2𝛿 (𝜃2)2𝛾 (2 + 𝑝)𝑠 𝛤(2𝛼 + 1) 𝛤(𝛼 + 1) 𝐴2 − 3 (𝜃1)𝛽 (𝜃3)𝛿 (𝜃2)𝛾 (1 + 𝑝)𝑠 𝐴 = 0 𝑎𝑡 𝑘 = 2, (𝜃1)4𝛽 (𝜃3)4𝛿 (𝜃2)4𝛾 (4 + 𝑝)𝑠 𝛤(4𝛼 + 1) 𝛤(2𝛼 + 1) 𝐴2 + (𝜃1)2𝛽 (𝜃3)2𝛿 (𝜃2)2𝛾 (3 + 𝑝)𝑠 𝛤(3𝛼 + 1) 𝛤(2𝛼 + 1) 𝐴1 − 3 (𝜃1)𝛽 (𝜃3)𝛿 (𝜃2)𝛾 (2 + 𝑝)𝑠 = 0 And so on. By equation (5) , we get following solution 𝑓(𝑤) = 1 𝑝𝑠 + 𝐵𝑤𝛼 + 1 𝛤(2𝛼 + 1) [ 3 𝑝𝑠 − 𝐵𝛤(𝛼 + 1)] + 1 𝛤(3𝛼 + 1) [− 3 𝑝𝑠 + 4𝐵𝛤(𝛼 + 1)] + 1 𝛤(4𝛼 + 1) [ 12 𝑝𝑠 − 5𝐵𝛤(𝛼 + 1)] + ⋯ 4. Conclusion We compute Caputo derivative formula of the Extended Hurwitz-Lerch Zeta function. Moreover, we obtained the solution of fractional differential equation involving Hurwitz-Lerch Zeta function. On the basis of the above result. We should be able to solve fractional differential equations involving other special functions, such as Mittag-Leffler functions, hypergeometric polynomials, and Jacobi polynomials. Acknowledgement: All authors would like to thanks integral University, Lucknow, India for providing the manuscript (MCN): IU/R&D/2024-MCN0003211 for this work. Conflict of interest: The authors declare that there is no conflict of interest. References [1] E.D.,Rainville,:Special Functions, The Macmillan Company, New York, 2013. [2] E. 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