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ExponentialSeries

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Description

Summary

A summation to +oo whose summand is a polynomial in the index times a power with
the index in the exponent, over a factorial of the index: sum(x^k / k!, k, 0, +oo) is e^x, and sum(3^(k+2) * (k^2 + k + 1) / (k + 3)!, k, 0, +oo) is
(8 * e^3 - 41) / 6.

Remarks

The whole family is e^c. Shift the index so the factorial is j!, and the
summand is q(j) * c^j / j! for a polynomial q; then
sum(j^n * c^j / j!, j, 0, +oo) = e^c * T_n(c), where T_n is the Touchard
polynomial sum(S(n, m) * c^m, m, 0, n) over the Stirling numbers of the second
kind -- j^n written in falling factorials, each of which sums to c^m * e^c.
A lower bound above the shifted zero subtracts the terms below it, which are finitely
many and exact. The series converges for every c, so no condition is owed.
 Recognised structurally, on the factors of the simplified summand: exactly one
(k + a)! ^ (-1) with a whole a, at most one c ^ (k + s) with
c free of the index, and a polynomial in the index for the rest; anything else
is left as written. A power whose exponent is not the index plus a constant --
c^(2k) -- is declined rather than rewritten, and so is a base that is zero.
The first of the orientation-week questions of
#1212.

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