N-Fractional calculus of products of some power functions and some doubly infinite sums

In this article, the N-fractional calculus of the following products of power functions:((z-c)α·(z-c)β)γ,((z-c)β·(z-c)α)γ, and((z-c)α+β)γ(z-c≠0) are discussed. Moreover, an application of the N-fractional calculus((z-c)β·(z-c)α)γ to the following doubly infinite sum:∑k=0∞∑m=0∞L(α,β,γ;k,m)-czkz-czm, whereL(α,β,γ;k,m)≔Γ(α+1)Γ(γ+1)Γ(k-α+m)Γ(γ-β-m)k!·m!Γ(α-k+1)Γ(γ-m+1)Γ(k-α)Γ(-β)(α,β,γ∉Z0+) is reported.