Describing the elementary theories of Rogers semilattices is one of the main problems of the theory of numberings. For the classical case of computable families of computably enumerable sets, V.V. V’jugin showed in [1] the existence of infinitely many families with pairwise elementarily different Rogers semilattices. Nevertheless, the study of Rogers semilattices undertaken in recent years (see...

Recently, the authors have established a large class of modular relations involving the Rogers-Ramanujan type functions J(q) andK(q) of order ten. In this paper, we establish further modular relations connecting these two functions with Rogers-Ramanujan functions, Göllnitz-Gordon functions and cubic functions, which are analogues to the Ramanujan’s forty identities for Rogers-Ramanujan function...

Rogers syndrome (Thiamine responsive megaloblastic anaemia-TRMA) occurs due to defect in the SLC19A2 gene. and SLC19A3 genes encode THTR1 2 respectively, which are thiamine transporter proteins. The gene is expressed inner ear cells, β-islet hematopoietic stem cells; consequently, typical clinical trial of TRMA diabetes, TRMA, sensorineural hearing loss. This syndrome, eponymously called Roger’...

We present two general finite extensions for each of the two Rogers-Ramanujan identities. Of these one can be derived directly from Watson’s transformation formula by specialization or through Bailey’s method, the second similar formula can be proved either by using the first formula and the q-Gosper algorithm, or through the so-called Bailey lattice.

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.