Partitions, $q$-polynomials and generating functions

I can answer part 2.

The co-efficient of $q^k$ in $\Psi_q(n)$ represents the number of elements greater than $k$ in all partitions of $n$

This can be proved with elementary analysis, mainly each part $k$ provides a contribution of $\{q^k,\dots,q^0\}$ to the q-nomial.

Part 1 is harder - both the differential of the partition function and the integral of the related inverted partition function are unresolved I believe.


A generating function of sorts is given by $$ \sum_{n\geq 1}\Psi_q(n)x^n = P(x)\sum_{m\geq 0}q^m\sum_{k\geq m+1} \frac{x^k}{1-x^k}, $$ where $P(x)=\prod_{i\geq 1}(1-x^i)^{-1}$.

Tags:

Partitions

Nt.Number Theory

Co.Combinatorics

Power Series

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