Papers

Let $\mathbb{K}$ be a non-Archimedean valued field. Let $n \in \mathbb{N}$. For every $(a_j)_{j=1}^n, (b_j)_{j=1}^n \in \mathbb{K}^n$, we show that \begin{align*} \left|\sum_{j=1}^{n}a_jb_j\right|^2\leq \max\left\{\left|\sum_{j=1}^n a_j^2\right|\left|\sum_{k=1}^nb^2_k\right|, \max_{1\leq j<k \leq n}|a_jb_k-a_kb_j|^2\right\}, \end{align*} \begin{align*} \left|\sum_{j=1}^n a_j^2\right|\left|\sum_{k=1}^nb^2_k\right|\leq \max\left\{\left|\sum_{j=1}^n a_jb_j\right|^2, \max_{1\leq j<k \leq n}|a_jb_k-a_kb_j|^2\right\}, \end{align*} \begin{align*} \left|AM(a_j)_{j=1}^n\right|\left|AM(b_j)_{j=1}^n\right|\leq \max\left\{ \left|AM(a_jb_j)_{j=1}^n\right|, \frac{1}{|n|^2}\max_{1\leq j < k \leq n}|a_j-a_k||b_j-b_k|\right\}, \end{align*} \begin{align*} \left|AM(a_jb_j)_{j=1}^n\right|\leq \max\left\{ \left|AM(a_j)_{j=1}^n\right|\left|AM(b_j)_{j=1}^n\right|, \frac{1}{|n|^2}\max_{1\leq j < k \leq n}|a_j-a_k||b_j-b_k|\right\}, \end{align*} where $AM$ denotes the arithmetic mean. First and second are non-Archimedean versions of Cauchy-Schwarz angle-length and third and fourth are Chebyshev arithmetic mean inequalities. Unlike in the Archimedean case, no conditions are required to derive non-Archimedean Chebyshev arithmetic mean inequalities.

Let $\mathbb{Z}^-=\setminus\{-1,-2,\dotsc\}$. In 2023, Qi and Lim gave two claims for summing the infinite series $$ \sum_{k=1}^{\infty} \binom{2k}{k} \frac{1}{\alpha+k} \biggl(\frac{\pm1}{4}\biggr)^k, \quad \alpha\in\mathbb{C}\setminus\mathbb{Z}^-. $$ In present paper, the authors establish several sum functions of the infinite and finite series $$ \sum_{k=1}^{\infty}\binom{2k}{k}\frac{1}{\alpha+k}\biggl(\frac{z}{4}\biggr)^k \quad\text{and}\quad \sum_{k=1}^{n}\binom{2k}{k}\frac{1}{\alpha+k}\biggl(\frac{z}{4}\biggr)^k $$ for $\alpha\in\mathbb{C}\setminus\mathbb{Z}^-$ and $n\in\mathbb{N}=\{1,2,\dotsc\}$ in terms of the Gauss hypergeometric functions ${}_2F_1$ and the generalized hypergeometric functions ${}_3F_2$ for $\alpha\in\mathbb{C}\setminus\mathbb{Z}^-$ and $n\in\mathbb{N}$. In light of the Euler integral representation of the Gauss hypergeometric function ${}_2F_1$, the author present several closed forms of two Gauss hypergeometric functions ${}_2F_1$, two generalized hypergeometric functions ${}_3F_2$, and the classical incomplete beta functions $B_z\bigl(\frac12, \frac{1}{2}+n\bigr)$ and $B_z\bigl(\frac12, 1+n\bigr)$. With the help of the Euler hypergeometric transform, the authors derive closed forms of five Gauss hypergeometric functions. In addition, the authors also obtain a closed form of the differential operator $\bigl[(1-z)\frac{\operatorname{d}}{\operatorname{d}z}(1-z)\bigr]^n \frac{\arcsin\sqrt{z}}{\sqrt{z(1-z)}}$ for $n\in\mathbb{N}_0=\{0\}\cup\mathbb{N}$.

The objective of this paper is to develop an extension of Kummer's second theorem and to establish generalized forms of four classical combinatorial identities—Knuth's old sum (also known as Reed–Dawson's combinatorial identity), Riordan's combinatorial identity, Gould's combinatorial identity, and Touchard's combinatorial identity—using a hypergeometric-series approach. Several new identities also arise as special cases of our main results.

We formalize the algebraic structure of rational polynomial phase sequences (discrete chirps) over the integer lattice $\mathbb{Z}$. By constructing a convolution algebra generated by polynomial phases with coefficients in $\mathbb{Q}$, we establish a rigorous operational calculus for discrete difference equations. We prove an exact orthogonality relation for these sequences over finite periodic windows, yielding a generalized discrete Fourier isometry. Furthermore, we demonstrate that the continuum limit of this discrete framework can be rigorously defined via the inductive limit of finite-dimensional $C^*$-algebras (an AF-algebra). This framework provides a rigorous algebraic substrate for finite-mode lattice approximations in quantum mechanics and the construction of mutually unbiased bases in finite-dimensional quantum information theory.