On the evaluation of sums of exponentiated multiples of generalized Catalan number linear combinations using a hypergeometric approach

Infinite series comprising exponentiated multiples of p-term linear combinations of Catalan numbers arise naturally from a related power series expansion for sin(2pα) (in odd powers of sin(α)) which itself has an interesting history. In this article some explicit results generated previously by the author (for p = 1, 2, 3) are discussed in the context of this general problem of series summation, and new evaluations made for the cases p = 4, 5 by way of further examples. A powerful hypergeometric approach is adopted which offers, from the analytical formulation developed, a means to achieve these particular evaluations and in principle many others for even greater values of p.

Infinite series comprising exponentiated multiples of p-term linear
combinations of Catalan numbers arise naturally from a related power
series expansion for sin(2pα) (in odd powers of sin(α)) which itself has
an interesting history. In this article some explicit results generated
previously by the author (for p = 1, 2, 3) are discussed in the context
of this general problem of series summation, and new evaluations
made for the cases p = 4, 5 by way of further examples. A powerful
hypergeometric approach is adopted which offers, from the analytical
formulation developed, a means to achieve these particular evaluations
and in principle many others for even greater values of p.