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

Abstract

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)} ,

Keywords

• the generalized harmonic number
• riordan arrays
• stirling number
• 11b99
• 11c20
• 15a23