  In this paper we propose a novel family of weighted orthonormal rational
functions on a semi-infinite interval. We write a sequence of
integer-coefficient polynomials in several forms and derive their corresponding
differential equations. These equations do not form Sturm-Liouville problems.
We overcome this disadvantage by multiplying some factors, resulting in a
sequence of irrational functions. We deduce various generating functions of
this sequence of irrational functions and find its associated Sturm-Liouville
problems, which brings orthogonality. Then we study a Hilbert space of
functions defined on a semi-infinite interval with its inner product induced by
a weight function determined by the Sturm-Liouville problems mentioned above.
We list two bases. One is the even subsequence of the irrational function
sequence above and another one is the non-positive integer power functions. We
raise one example of Fourier series expansion and one example of interpolation
as applications.
