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PolynomialGeometricSeries

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Description

Summary

A summation whose summand is a polynomial in the index times a power with the index in
the exponent: sum(k 2^k, k, 1, n) is (n - 1) 2^(n + 1) + 2, and
sum((-1)^(k - 1) k^2, k, 1, n) is (-1)^(n - 1) n (n + 1)/2, which is
§5.3.4 Try 6 of Sullivan and Mackey's An Introduction to Proofs.

Remarks

The summand is read as p(k) C b^(m k + s), which is C b^s p(k) r^k with the
ratio r = b^m, the way GeometricSeries reads its summand, with a
polynomial p of degree at least one beside the power. For a ratio other than 1
there is a polynomial q of the same degree with q(k + 1) r - q(k) = p(k),
so that q(k) r^k is a discrete antiderivative of the summand, and the sum from
a to b is q(b + 1) r^(b + 1) - q(a) r^a. The coefficients of
q are found highest first: the coefficient of k^j in q(k + 1) r - q(k) is (r - 1) q_j + r sum_{i > j} binomial(i, j) q_i, one division by r - 1 each, which is the whole of the linear algebra -- Gosper's algorithm specialised to a
summand whose ratio of consecutive terms is a rational function with a constant
denominator.
 At r = 1 the summand is the polynomial, which PolynomialSummation answers; a symbolic ratio carries provided not r = 1, since the formula divides
by r - 1. To +oo the series converges exactly when |r| < 1, to
-q(a) r^a, since q(k) r^k then goes to zero. Part of
#1409.

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