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Register a new userA Five-variable generalization of Ramanujans reciprocity theorem
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Xinrong Ma
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By virtue of Baileys well-known bilateral 6psi_6 summation formula and Watsons transformation formula,we extend the four-variable generalization of Ramanujans reciprocity theorem due to Andrews to a five-variable one. Some relevant new q-series identities including a new proof of Ramanujans reciprocity theorem and of Watsons quintuple product identity only based on Jacksons transformation are presented.
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Let $n$ be a positive integer. In 1915, Theisinger proved that if $nge 2$, then the $n$-th harmonic sum $sum_{k=1}^nfrac{1}{k}$ is not an integer. Let $a$ and $b$ be positive integers. In 1923, Nagell extended Theisingers theorem by showing that the reciprocal sum $sum_{k=1}^{n}frac{1}{a+(k-1)b}$ is not an integer if $nge 2$. In 1946, ErdH{o}s and Niven proved a theorem of a similar nature that states that there is only a finite number of integers $n$ for which one or more of the elementary symmetric functions of $1,1/2, ..., 1/n$ is an integer. In this paper, we present a generalization of Nagells theorem. In fact, we show that for arbitrary $n$ positive integers $s_1, ..., s_n$ (not necessarily distinct and not necessarily monotonic), the following reciprocal power sum $$sumlimits_{k=1}^{n}frac{1}{(a+(k-1)b)^{s_{k}}}$$ is never an integer if $nge 2$. The proof of our result is analytic and $p$-adic in character.
Recently, Corvaja and Zannier obtained an extension of the Subspace Theorem with arbitrary homogeneous polynomials of arbitrary degreee instead of linear forms. Their result states that the set of solutions in P^n(K) (K number field) of the inequality being considered is not Zariski dense. In our paper we prove by a different method a generalization of their result, in which the solutions are taken from an arbitrary projective variety X instead of P^n. Further, we give a quantitative version which states in a precise form that the solutions with large height lie ina finite number of proper subvarieties of X, with explicit upper bounds for the number and for the degrees of these subvarieties.
Let $G$ be a connected reductive group over a $p$-adic local field $F$. We propose and study the notions of $G$-$varphi$-modules and $G$-$(varphi, abla)$-modules over the Robba ring, which are exact faithful $F$-linear tensor functors from the category of $G$-representations on finite-dimensional $F$-vector spaces to the categories of $varphi$-modules and $(varphi, abla)$-modules over the Robba ring, respectively, commuting with the respective fiber functors. We study Kedlayas slope filtration theorem in this context, and show that $G$-$(varphi, abla)$-modules over the Robba ring are $G$-quasi-unipotent, which is a generalization of the $p$-adic local monodromy theorem proven independently by Y. Andre, K. S. Kedlaya, and Z. Mebkhout.
In 1730 James Stirling, building on the work of Abraham de Moivre, published what is known as Stirlings approximation of $n!$. He gave a good formula which is asymptotic to $n!$. Since then hundreds of papers have given alternative proofs of his result and improved upon it, including notably by Burside, Gosper, and Mortici. However Srinivasa Ramanujan gave a remarkably better asymptotic formula. Hirschhorn and Villarino gave a nice proof of Ramanujans result and an error estimate for the approximation. In recent years there have been several improvements of Stirlings formula including by Nemes, Windschitl, and Chen. Here it is shown (i) how all these asymptotic results can be easily verified; (ii) how Hirschhorn and Villarinos argument allows a tweaking of Ramanujans result to give a better approximation; (iii) that a new asymptotic formula can be obtained by further tweaking of Ramanujans result; (iv) that Chens asymptotic formula is better than the others mentioned here, and the new asymptotic formula is comparable with Chens.
We define a new parameter $A_{k,n}$ involving Ramanujans theta-functions for any positive real numbers $k$ and $n$ which is analogous to the parameter $A_{k,n}$ defined by Nipen Saikia cite{NS1}. We establish some modular relation involving $A_{k,n}$ and $A_{k,n}$ to find some explicit values of $A_{k,n}$. We use these parameters to establish few general theorems for explicit evaluations of ratios of theta functions involving $varphi(q)$.
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