This paper is concerned with the system of simultaneous diophantine equations $\sum_{i=1}^6A_i^k=\sum_{i=1}^6B_i^k$ for $k=2, 4, 6, 8, 10.$ Till now only two numerical solutions of the system are known. This paper provides an infinite family of solutions. It is well-known that solutions of the above system lead to ideal solutions of the Tarry-Escott Problem of degree $11$, that is, of the system of simultaneous equations, $\sum_{i=1}^{12}a_i^k=\sum_{i=1}^{12}b_i^k$ for $k=1, 2, 3,\ldots,11.$ We use one of the ideal solutions to prove new results on sums of $13^{th}$ powers. In particular, we prove that every integer can be expressed as a sum or difference of at most $27$ thirteenth powers of positive integers.
Using Borwein's simple analytic method for the irrationality of the $q$-logarithm at rational points, we prove a quite general result on arithmetic properties of certain series, where the entering parameters are algebraic numbers. More precisely, our main result says that $\sum_{k\ge1}\beta^k/(1-\alpha q^k)$ is not in $\mathbb{Q}(q)$, if $q$ is an algebraic integer with all its conjugates (if any) in the open unit disc, if $\alpha\in\mathbb{Q}(q)^\times\setminus\{q^{-1},q^{-2},\ldots\}$ satisfies a mild denominator condition (implying $|q|>1$), and if $\beta$ is a unit in $\mathbb{Q}(q)$ with $|\beta|\le1$ but no other conjugates in the open unit disc. Our applications concern meromorphic functions defined in $|z|<|u|^{a\ell}$ by power series $\sum_{n\ge1}z^n/(\prod_{0\le\lambda<\ell}R_{a(n+\lambda)+b})$, where $R_m:=gu^m+hv^m$ with non-zero $u,v,g,h$ satisfying $|u|>|v|, R_m\ne0$ for any $m\ge1$, and $a,b+1,\ell$ are positive rational integers. Clearly, the case where $R_m$ are the Fibonacci or Lucas numbers is of particular interest. It should be noted that power series of the above type were first studied by Wynn from the analytical point of view.
The Kronecker limit formula for a positive definite binary quadratic form or the Dedekind zeta-function of an imaginary quadratic field is quite well-known and there exists an enormous amount of literature pertaining to its proof and applications. Here we give a different kind of proof depending on the hypergeometric transform. The idea goes back to Koshilyakov and we adopted it in this note to give a new derivation of the formula. Here the connection formula for the hypergeometric function plays an essential role.