Please rewrite your expression and your question clearly, ...

Is P(n,r) stands for n permutation r. then you need to use umbrall calculus see the book "Introduction to Combinatorial Analysis - John Riordan "

As I understand it, you are asking for an expression with respect to the number of permutations of n items taken r at a time; i.e., your index should be r instead of k because otherwise it is not bound in the expression ∑ⁿᵣ₌₀P(n,r) xʳ a⁽ⁿ⁻ʳ⁾ that is analogous the Binomial Theorem (x+a)ⁿ = ∑ⁿᵣ₌₀ (ⁿᵣ) xʳ a⁽ⁿ⁻ʳ⁾. (I'd love to be able to use Mathjax to present a better expression, but Unicode will have to do for now.)

P(n,r) = (ⁿᵣ) r! = (n)ᵣ, also known as Pochhammer's symbol, or falling factorial is the coefficient of each term xʳ a⁽ⁿ⁻ʳ⁾. Hence, (x+a)⁽ⁿ⁾ is the simplest closed-form expression that I can think of, where the exponent n is replaced by (n), with the subscript r removed (and understood as the falling factorial in context), as it is the index for the coefficients P(n,r).

The first four polynomials are:

(x+a)⁽¹⁾ = (1)₀x+(1)₁a = x + a

(x+a)⁽²⁾ = (2)₀x²+(2)₁ax+(2)₂a² = x² + 2ax + 2a²

(x+a)⁽³⁾ = (3)₀x³+(3)₁ax²+(3)₂a²x+(3)₃a³ = x³ + 3ax² + 6a²x + 6a³

(x+a)⁽⁴⁾ = (4)₀x⁴+(4)₁ax³+(4)₂a²x²+(4)₃a³x+(4)₄a⁴ = x⁴ + 4ax³ + 12a²x² + 24a³x + 24a⁴

BTW, the Umbral Taylor series would give the same expression, but the overarching theory that abstracts all of the above would probably be related to generalized hypergeometric, Fox-Wright, and elliptic hypergeometric functions. If the recent proof of the ABC conjecture by Mochizuki holds, then we can expect much deeper connections to surface.

- Marshall Mayberry