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In this paper, we explore the role that Liu's transformation formula can play in discovering Rogers-Ramanujan type identities. Specifically, we combine Liu's transformation formula with other $q$-series summations to derive a series of parameterized identities. Thereafter, through careful selection of these parameters, we obtain $75$ Rogers-Ramanujan type identities from Slater's list, and uncover several new Rogers-Ramanujan type identities.

The primary purpose of this paper is to provide a survey of properties, values, identities, and generalizations of the Rogers--Ramanujan continued fraction, which is closely related to the Rogers--Ramanujan identities. Many of these results are found in Ramanujan's first two letters to Hardy, Ramanujan's notebooks, and his lost notebook. Short historical accounts are provided for both the notebooks and lost notebook.

We establish some new bilateral double-sum Rogers-Ramanujan identities involving parameters. As applications, these identities yield several new multi-sum Rogers-Ramanujan type identities. Our proofs utilize the theory of basic hypergeometric series in conjunction with the integral method.

We note that an argument by Rogers (1958) gives a proof of Vaaler's theorem (1979) about sections of the cube and allows certain generalizations of the theorem.

We prove four new Rogers-Ramanujan-type identities for double series. They follow from the classical Rogers-Ramanujan identities using the constant term method and properties of Rogers-Szegő polynomials.

The Rogers-Ramanujan identities are investigated using the Cauchy identity for Schur functions.

Roger Carter (1934--2022) was a very well known mathematician working in algebra, representation theory and Lie theory. He spent most of his mathematical career in Warwick. Roger was a great communicator of mathematics: the clarity, precision and enthusiasm of his lectures delivered in his beautiful handwriting were hallmark features recalled by numerous students and colleagues. His books have been described as marvelous pieces of scholarship and service to the general mathematical community. We both met Roger early in our careers, and were encouraged and influenced by him~ -- ~and his lovely sense of humour. This text is our tribute, both to his mathematical achievements, and to his kindness and generosity towards his students, his colleagues, his collaborators, and his family.

In 2010, Andrews considers a variety of parity questions connected to classical partition identities of Euler, Rogers, Ramanujan and Gordon. As a large part in his paper, Andrews considered the partitions by restricting the parity of occurrences of even numbers or odd numbers in the Rogers-Ramanujan-Gordon type. The Rogers-Ramanujan-Gordon type partition was defined by Gordon in 1961 as a combinatorial generalization of the Rogers-Ramaujan identities with odd moduli. In 1974, Andrews derived an identity which can be considered as the generating function counterpart of the Rogers-Ramanujan-Gordon theorem, and since then it has been called the Andrews--Gordon identity. By revisting the Andrews--Gordon identity Andrews extended his results by considering some additional restrictions involving parities to obtain some Rogers-Ramanujan-Gordon type theorems and Andrews--Gordon type identities. In the end of Andrews' paper, he posed $15$ open problems. Most of Andrews' $15$ open problems have been settled, but the $11$th that "extend the parity indices to overpartitions in a manner" has not. In 2013, Chen, Sang and Shi, derived the overpartition analogues of the Rogers-Ramanujan-Gordon the

By the twelfth century, northern European scholars gradually embraced Arabic innovations in science and technology. England naturally developed into a significant centre of the new learning in western Europe. Hereford, and specifically its cathedral school, played a particularly important role in the transition of English scholarship to the new learning. Hereford cathedral developed into a focal point for high-level scholarship, attracting numerous scholars from across the continent. Roger of Hereford stands out among his peers as an enlightened scholar who made more practical use than most of the full astronomical and astrological knowledge base available in England at the time. A significant body of recent scholarship focuses on twelfth-century ecclesiastical developments, including those relating to Roger of Hereford's Computus. However, much less scholarly emphasis is placed on Roger's astronomical calculations, particularly those which allowed him to establish an important reference meridian at Hereford. Those aspects are the focus of this paper.

Ce texte, écrit pour la Gazette de la Société mathématique de France, évoque les fonctions de type positif et leur histoire avant 1950 ; on y présente notamment des extraits de lettres écrites par Roger Godement, qui leur consacra sa thèse en 1946. This is an expository paper on the early history of positive-definite functions, written for the "Gazette de la Société Mathématique de France". It contains pictures of letters written by Roger Godement, during and after the preparation of his 1946 thesis about positive-definite functions on groups.

We give new proofs of the twelve Rogers-Ramanujan-type identities due to Rogers and Slater that are traditionally associated with the moduli 7, 14 and 28.

We generalize the positivity conjecture on (Kauffman bracket) skein algebras to Roger--Yang skein algebras. To generalize it, we use explicit polynomials like Chebyshev polynomials of the first kind to give candidates of positive bases. Moreover, the polynomials form a lower bound in the sense of [Lê18] and [LTY21]. We also discuss a relation between the polynomials and the centers of Roger--Yang skein algebras when the quantum parameter is a complex root of unity.

We present an operator approach to deriving Mehler's formula and the Rogers formula for the bivariate Rogers-Szegö polynomials $h_n(x,y|q)$. The proof of Mehler's formula can be considered as a new approach to the nonsymmetric Poisson kernel formula for the continuous big $q$-Hermite polynomials $H_n(x;a|q)$ due to Askey, Rahman and Suslov. Mehler's formula for $h_n(x,y|q)$ involves a ${}_3φ_2$ sum and the Rogers formula involves a ${}_2φ_1$ sum. The proofs of these results are based on parameter augmentation with respect to the $q$-exponential operator and the homogeneous $q$-shift operator in two variables. By extending recent results on the Rogers-Szegö polynomials $h_n(x|q)$ due to Hou, Lascoux and Mu, we obtain another Rogers-type formula for $h_n(x,y|q)$. Finally, we give a change of base formula for $H_n(x;a|q)$ which can be used to evaluate some integrals by using the Askey-Wilson integral.

The two Rogers-Ramanujan $q$-series \[ \sum_{n=0}^{\infty}\frac{q^{n(n+σ)}}{(1-q)\cdots (1-q^n)}, \] where $σ=0,1$, play many roles in mathematics and physics. By the Rogers-Ramanujan identities, they are essentially modular functions. Their quotient, the Rogers-Ramanujan continued fraction, has the special property that its singular values are algebraic integral units. We find a framework which extends the Rogers-Ramanujan identities to doubly-infinite families of $q$-series identities. If $a\in\{1,2\}$ and $m,n\geq 1$, then we have \[ \sum_{\substack{λλ_1\leq m}} q^{a|λ|} P_{2λ}(1,q,q^2,\dots;q^n) =\textrm{"infinite product modular function"}, \] where the $P_λ(x_1,x_2,\dots;q)$ are Hall-Littlewood polynomials. These $q$-series are specialized characters of affine Kac--Moody algebras. Generalizing the Rogers-Ramanujan continued fraction, we prove in the case of $\textrm{A}_{2n}^{(2)}$ that the relevant $q$-series quotients are integral units.

The aim of this paper is to study the convergence and divergence of the Rogers-Ramanujan and the generalized Rogers-Ramanujan continued fractions on the unit circle. We provide an example of an uncountable set of measure zero on which the Rogers-Ramanujan continued fraction $R(x)$ diverges and which enlarges a set previously found by Bowman and Mc Laughlin. We further study the generalized Rogers-Ramanujan continued fractions $R_a(x)$ for roots of unity $a$ and give explicit convergence and divergence conditions. As such, we extend some work of Huang towards a question originally investigated by Ramanujan and some work of Schur on the convergence of $R(x)$ at roots of unity. In the end, we state several conjectures and possible directions for generalizing Schur's result to all Rogers-Ramanujan continued fractions $R_a(x)$.

We present a new proof of the Rogers-Ramanujan identities. Surprisingly, all its ingredients are available already in Rogers seminal paper from 1894, where he gave a considerably more complicated proof.

A numbering of a countable family $S$ is a surjective map from the set of natural numbers $ω$ onto $S$. A numbering $ν$ is reducible to a numbering $μ$ if there is an effective procedure which given a $ν$-index of an object from $S$, computes a $μ$-index for the same object. The reducibility between numberings gives rise to a class of upper semilattices, which are usually called Rogers semilattices. The paper studies Rogers semilattices for families $S \subset P(ω)$ belonging to various levels of the analytical hierarchy. We prove that for any non-zero natural numbers $m eq n$, any non-trivial Rogers semilattice of a $Π^1_m$-computable family cannot be isomorphic to a Rogers semilattice of a $Π^1_n$-computable family. One of the key ingredients of the proof is an application of the result by Downey and Knight on degree spectra of linear orders.

By studying non-commutative series in an infinite alphabet we introduce shift-plethystic trees and a class of integer compositions as new combinatorial models for the Rogers-Ramanujan identities. We prove that the language associated to shift-plethystic trees can be expressed as a non-commutative generalization of the Rogers-Ramanujan continued fraction. By specializing the noncommutative series to $q$-series we obtain new combinatorial interpretations to the Rogers-Ramanujan identities in terms of signed integer compositions. We introduce the operation of shift-plethysm on non-commutative series and use this to obtain interesting enumerative identities involving compositions and partitions related to Rogers-Ramanujan identities.

A generalized Bailey pair, which contains several special cases considered by Bailey (\emph{Proc. London Math. Soc. (2)}, 50 (1949), 421--435), is derived and used to find a number of new Rogers-Ramanujan type identities. Consideration of associated $q$-difference equations points to a connection with a mild extension of Gordon's combinatorial generalization of the Rogers-Ramanujan identities (\emph{Amer. J. Math.}, 83 (1961), 393--399). This, in turn, allows the formulation of natural combinatorial interpretations of many of the identities in Slater's list (\emph{Proc. London Math. Soc. (2)} 54 (1952), 147--167), as well as the new identities presented here. A list of 26 new double sum--product Rogers-Ramanujan type identities are included as an appendix.