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Let $f(n)$ be an arbitrary function with integer values. Let $a(n)$ be an integer sequence such that $$ \frac{1}{1-x}=\sum\limits_{n=0}^{\infty}a(n)x^n\prod\limits_{k=1}^{n+1}(1-f(k)x) $$ Start with ...

In the 1988 Narosa edition of Ramanujan's The Lost Notebook and Other Unpublished Papers, on the first line of page 1 is the following: $$ \Big(1+\frac1a\Big) \Bigg\{\frac{1}{(1-aq)(1-q/a)}+\frac{q(1+...

This question arose from the recent one, roots of a polynomial linked to mock theta function?. Let $$ g(x):=\sum_{k=0}^\infty x^k\prod_{j=1}^{k-1}(1 + x^j)^2\\=1+x+x^2+3 x^3+4 x^4+6 x^5+10 x^6+15 x^7+...

The Fibonacci numbers $F_n$ can be given by $$\sum_{k\geq0}F_kx^k=\frac{x}{1-x-x^2}.$$ Among many many properties of this sequence, consider the following two results: (1) the coefficients of the ...