A harmonic M-factorial function and applications

We offer analogs to the falling factorial and rising factorial functions for the set of harmonic numbers, as well as a mixed factorial function called the M-factorial. From these concepts, we develop a harmonic analog of the binomial coefficient and an alternate expression of the harmonic exponential function and establish several identities. We generalize from the harmonic numbers to a general time scale and demonstrate how solutions to some second order eigenvalue problems and partial dynamic equations can be constructed using power series built from the M-factorial function --Abstract, page iii

On identities involving generalized harmonic, hyperharmonic and special numbers with Riordan arrays

In this paper, by means of the summation property to the Riordan array, we derive some identities involving generalized harmonic, hyperharmonic and special numbers. For example, for n ≥ 0,∑k=0nBkk!H(n.k,α)=αH(n+1,1,α)-H(n,1,α),\sum\limits_{k = 0}^n {{{{B_k}} \over {k!}}H\left( {n.k,\alpha } \right) = \alpha H\left( {n + 1,1,\alpha } \right) - H\left( {n,1,\alpha } \right)} ,and for n > r ≥ 0, ∑k=rn-1(-1)ks(k,r)r!αkk!Hn-k(α)=(-1)rH(n,r,α),\sum\limits_{k = r}^{n - 1} {{{\left( { - 1} \right)}^k}{{s\left( {k,r} \right)r!} \over {{\alpha ^k}k!}}{H_{n - k}}\left( \alpha \right) = {{\left( { - 1} \right)}^r}H\left( {n,r,\alpha } \right)}