Generating functions for series involving higher powers of inverse
binomial coefficients and their applications
The purpose of this paper is to construct generating functions in terms
of hypergeometric function and logarithm function for finite and
infinite sums involving higher powers of inverse binomial coefficients.
These generating functions provide a novel way of examining higher
powers of inverse binomial coefficients from the perspective of these
sums, assessing how several of these sums and these coefficients related
to each other. A unique relation between the Euler-Frobenius polynomial
and B-spline associated with exponential Euler spline is reported.
Moreover, with the aid of derivative operator and functional equations
for generating functions, many new computational formulas involving the
special finite sums of higher powers of (inverse) binomial coefficients,
the Bernoulli polynomials and numbers, Euler polynomials and numbers,
the Stirling numbers, the harmonic numbers, and finite sums are derived.
Moreover, A few recurrence relations containing these particular finite
sums are given. Using these recurence relations, we give a solution of
the problem which was given by Charalambides7, Exercise 30, p. 273. We
give calculations algorithms for these finite sums. Applying these
algorithms and Wolfram Mathematica 12.0, we give some plots and many
values of these polynomials and finite sums.