Wallis-Ramanujan-Schur-Feynman

Schur ' s Determinants and Partition

Garrett, Ismail, and Stanton gave a general formula that contains the Rogers{ Ramanujan identities as special cases. The theory of associated orthogonal polynomials is then used to explain determinants that Schur introduced in 1917 and show that the Rogers{Ramanujan identities imply the Garrett, Ismail, and Stanton seemingly more general formula. Using a result of Slater a continued fraction is...

Math 7012 Enumerative Combinatorics Project: Introduction to a combinatorial proof of the Rogers-Ramanujan and Schur identities and an application of Rogers-Ramanujan identity

Partial-sum Analogues of the Rogers–ramanujan Identities

A new polynomial analogue of the Rogers–Ramanujan identities is proven. Here the product-side of the Rogers–Ramanujan identities is replaced by a partial theta sum and the sum-side by a weighted sum over Schur polynomials.

Partial-Sum Analogues of the Rogers - Ramanujan Identities

A new type of polynomial analogue of the Rogers–Ramanujan identities is proven. Here the product-side of the Rogers–Ramanujan identities is replaced by a partial theta sum and the sum-side by a weighted sum over Schur polynomials.

A combinatorial proof of the Rogers-Ramanujan and Schur identities

We give a combinatorial proof of the first Rogers-Ramanujan identity by using two symmetries of a new generalization of Dyson’s rank. These symmetries are established by direct bijections.