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The polynomials calculating sums of powers of arithmetic progressions are polynomials in a variable that depend both on the particular arithmetic progression constituting the basis of the summed powers and on the constant exponent, non-negative integer, chosen. Their degree always exceeds the constant exponent by one unit and have the property that when the polynomial variable coincides with the number of summed addends, the result of the polynomial function also coincides with that of the sum.
The problem therefore consists in finding S h , d m {\displaystyle S_{h,d}^{m}} i.e. polynomials as a function of n {\displaystyle n} calculating sums of n {\displaystyle n} addends:
with m {\displaystyle m} and n {\displaystyle n} integers positive, h {\displaystyle h} first term of a arithmetic progression and d ≠ 0 {\displaystyle d\neq 0} the common difference.The two parameters can be not only integers but also rational, real and even complex.