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Page 27 of Ramanujan's Lost Notebook contains a beautiful identity which not only gives, as a special case, a famous modular relation between the Rogers-Ramanujan functions $G(q)$ and $H(q)$ but also a relation between two fifth order mock…
Quantum theory was discovered in an adventurous way, under the urge to solve puzzles-like the spectrum of the blackbody radiation-that haunted the physics community at the beginning of the 20th century. It soon became clear, though, that…
New status in quantum mechanics is connected with recent achievements in the inverse problem. With its help instead of about ten exactly solvable models which serve as a basis of the contemporary education there are infinite (!) number,…
By virtue of Bailey's well-known bilateral 6\psi_6 summation formula and Watson's transformation formula,we extend the four-variable generalization of Ramanujan's reciprocity theorem due to Andrews to a five-variable one. Some relevant new…
Deutsch, Feynman, and Manin viewed quantum computing as a kind of universal physical simulation procedure. Much of the writing about quantum Turing machines has shown how these machines can simulate an arbitrary unitary transformation on a…
We discuss fundamentals of quantum computing and information - quantum gates, circuits, algorithms, theorems, error correction, and provide collection of QISKIT programs and exercises for the interested reader.
In this lecture notes we try to familiarize the audience with the theory of Bernoulli polynomials; we study their properties, and we give, with proofs and references, some of the most relevant results related to them. Several applications…
In a handwritten manuscript published with his lost notebook, Ramanujan stated without proofs forty identities for the Rogers-Ramanujan functions. We observe that the functions that appear in Ramanujan's identities can be obtained from a…
Ramanujan recorded five $q$-series identities at the end of his second notebook and an unified generalization of these identities obtained by Bhoria, Eyyunni and Maji. Recently, Dixit and Patel gave a finite analogue of the identity of…
Generalizing a sequence of Lambert, Cayley and Ramanujan, Chapoton has recently introduced a polynomial sequence Q_n:=Q_n(x,y,z,t) defined by Q_1=1, Q_{n+1}=[x+nz+(y+t)(n+y\partial_y)]Q_n. In this paper we prove Chapoton's conjecture on the…
Prof. K.G Ramanathan was a legendary Indian Mathematician, working in Number Theory and a prolific Institution builder. Apart from this, he was an excellent teacher and influenced several brilliant students. In this article, we overview his…
We provide an historical account of equivalent conditions for the Riemann Hypothesis arising from the work of Ramanujan and, later, Guy Robin on generalized highly composite numbers. The first part of the paper is on the mathematical…
In the last few years, theoretical study of quantum systems serving as computational devices has achieved tremendous progress. We now have strong theoretical evidence that quantum computers, if built, might be used as a dramatically…
Let $\beta$ be a positive integer. A generalization of the Ramanujan sum due to Cohen is given by \begin{align} c_{q,\beta }(n) := \sum\limits_{{{(h,{q^\beta })}_\beta } = 1} {{e^{2\pi inh/{q^\beta }}}}, \nonumber \end{align} where $h$…
It is argued that the conclusions obtained by Renninger (Zeitschrift fur Physik 136, 251 (1953)), by means of an interferometer thought experiment, have important implications for a number of still ongoing discussions about quantum…
Situated as a language between computer science, quantum physics and mathematics, tensor network theory has steadily grown in popularity and can now be found in applications ranging across the entire field of quantum information processing.…
Starting with Ramanujan's famous taxicab problem, we can study the solvability of the equations $p^n+q^n=r^n+s^n$ and, more generally, $p_1^{k_1}+\dots+p_m^{k_m}=0$ among polynomials.
We develop the work of Christandl et al. [M. Christandl, N. Datta, T. C. Dorlas, A. Ekert, A. Kay, and A. J. Landahl, Phys. Rev. A 71, 032312 (2005)], to show how a d-hypercube homogenous network can be dressed by additional links to…
Ramanujan presented four identities for third order mock theta functions in his Lost Notebook. In 2005, with the aid of complex analysis, Yesilyurt first proved these four identities. Recently, Andrews et al. provided different proofs by…
It is von Neumann who opened the window for today's Information epoch. He defined quantum entropy including Shannon's information more than 20 years ahead of Shannon, and he introduced a concept what computation means mathematically. In…