Abstract
We study the following family of integral-valued alternating sums, where 
and are integers:
We first consider
for
and
non-negative integers and show that it is of the form
, where
may be represented as a polynomial of degree
in
, or expressed as a non-polynomial closed form given by a sum of binomial numbers. We then consider
for
a negative integer and for
a non-negative integer. This reveals, in particular, that
for
, that
for
. We also show that
is a polynomial of degree
in
, for fixed
, with
, and we give expressions for the coefficients.
Acknowledgements
The author wishes to thank an anonymous referee for their very constructive suggestions relating to the cases where .
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