Some debts I owe
This is an unusual article with detailed descriptions of the author's own mathematical experience. The author relates the beginnings of the paths leading to his work on Ramanujan's Lost Notebook, Bailey Chains, Determinant Evaluations and Partition Analysis. In the final section of this article, the author describes the following interesting problem: Let \[ L_k(q) = \sum_{n_1,n_2,\dots,n_{k+1}\geq 1}q^{n_1n_2+n_2n_3+\cdots +n_kn_{k+1}}, \] and \[ M_k(q) = \sum_{n=1}^\infty \frac{(-1)^nq^{n(n+1)/2}}{(1-q)(1-q^2)\cdots (1-q^{n-1})(1-q^n)^{k+1}}. \] For \(k=1,2\) and \(3\), one has \[ M_1(q)=L_1(q), \] \[ M_2(q) = L_1(q)+L_2(q)+L_3(q), \] and \[ M_3(q) = L_1(q)+2L_2(q)+3L_3(q)+2L_4(q)+L_5(q). \] The author ``challenges'' the reader to find the right pattern of the above relations for \(k>3.\) He refers to this as ``the Liouville mystery''. Besides the interesting stories and mathematical results, there is a word of wisdom which stands out in this article. It was from Marco Schützenberger and went as follow: Do not be too concerned about your current disappointment. The best that any mathematician can really hope for is to prove some first class theorems and have them understood and appreciated by a few good mathematicians. Everything else is politics.
scientific article; zbMATH DE number 1301083