Fix a positive integer $k$. Let $b_n(q)$ be the polynomial defined by the recursive equation $$b_n(q)=\binom{n+k-1}k+(1+q^{n-1})b_{n-1}(q), \qquad n\geq1,$$ initializing with $b_0(q)=0$.

I run into the below through experiment.

QUESTION. Denote $(q)_m=(1-q)(1-q^2)\cdots(1-q^m)$. Is this true? $$\lim_{n\rightarrow\infty}\left(b_n(q)-\binom{n+k}{k+1}\right) =\sum_{m=1}^{\infty}\frac{q^{\binom{m+1}2}}{(q)_m(1-q^m)^{k+1}}.$$

Fix an integer $k$. Let $b_n(q)$ be the polynomial defined by the recursive equation $$b_n(q)=\binom{n+k-1}k+(1+q^{n-1})b_{n-1}(q), \qquad n\geq1,$$ initializing with $b_0(q)=0$.

I run into the below through experiment.

QUESTION. Denote $(q)_m=(1-q)(1-q^2)\cdots(1-q^m)$. Is this true? $$\lim_{n\rightarrow\infty}\left(b_n(q)-\binom{n+k}{k+1}\right) =\sum_{m=1}^{\infty}\frac{q^{\binom{m+1}2}}{(q)_m(1-q^m)^{k+1}}.$$

NOTE. The case $k=-1$ recovers the generating function for partitions into distinct parts.

Fix a positive integer $k$. Let $b_n(q)$ be the polynomial defined by the recursive equation $$b_n(q)=\binom{n+k-1}k+(1+q^{n-1})b_{n-1}(q), \qquad n\geq1,$$ initializing with $b_0(q)=0$.

I run into the below through experiment.

QUESTION. Is this true? $$\lim_{n\rightarrow\infty}\left(b_n(q)-\binom{n+k}{k+1}\right) =\sum_{m=1}^{\infty}\frac{q^{\binom{m+1}2}}{(q)_m(1-q^m)^{k+1}}.$$

Fix a positive integer $k$. Let $b_n(q)$ be the polynomial defined by the recursive equation $$b_n(q)=\binom{n+k-1}k+(1+q^{n-1})b_{n-1}(q), \qquad n\geq1,$$ initializing with $b_0(q)=0$.

I run into the below through experiment.

QUESTION. Denote $(q)_m=(1-q)(1-q^2)\cdots(1-q^m)$. Is this true? $$\lim_{n\rightarrow\infty}\left(b_n(q)-\binom{n+k}{k+1}\right) =\sum_{m=1}^{\infty}\frac{q^{\binom{m+1}2}}{(q)_m(1-q^m)^{k+1}}.$$