The discrete hypergeometric series ${}_pF_q$ defined in this paper is $$ {}_pF_q(a_1,\ldots,a_p;b_1,\ldots,b_q;t,n,\xi) = \displaystyle\sum_{k=0}^{\infty} \dfrac{(a_1)_k\ldots(a_p)_k}{(b_1)_k\ldots(b_q)_k} \dfrac{t^{\underline{k}}}{k!},$$ where $(x)_k$ denotes a so-called "Pochhammer symbol.

The classical Chebyshev polynomials (of the first kind) solve the differential equation \[ (1-t^2)y''-ty'+n^2y=0\] and have hypergeometric representation \[ \mathscr{T}_n(t) = {}_2F_1 \left( -n, n; \dfrac{1}{2}, \dfrac{1}{2}(1-t) \right).\]

The traditional sine-integral function $\mathscr{Si}$ is defined by $\mathscr{Si}(t)=\displaystyle\int_0^t \dfrac{\sin(t)}{t} \mathrm{d}t$. It obeys the differential equation $$ty^{\prime \prime \prime}+2y^{\prime \prime}+ty=0,$$ and it obeys the power series $$\mathscr{Si}(t) = \displaystyle\sum_{k=1}^{\infty} \dfrac{(-1)^{k-1}t^{2k-1}}{(2k-1)(2k-1)!}.$$