A NOTE ON A GENERALIZED M-SERIES AS A SPECIAL FUNCTION OF FRACTIONAL CALCULUS

By Manoj Sharma, R. Jain. 2009

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A generalization of truncated M-fractional derivative and applications to fractional differential equations (2020, background): “In 2009, generalized M-series defined by Sharma and Jain [25, 26] β ,γ pMq (z) := β ,γ pMq [ a1 · · · ap c1 · · · cq ;z ] = ∞ ∑ k=0 (a1)k · · ·(ap)k (c1)k · · ·(cq)k zk Γ(βk+ γ) where β ,γ,z ∈ C, p,q ∈ N, ℜ(β ) > 0, ci 6= 0,−1,−2, . . .(i = 1,2, . . . ,q).”
Integral and computational representations of the extended Hurwitz–Lerch zeta function (2011, background): “2) reduces to the following generalized M-series which was recently introduced by Sharma and Jain [27]:”