相关论文: A set of the Vi\`ete-like recurrence relations for…

We present a dispersive analysis of the decay amplitude for $\eta'\to\eta\pi\pi$ that is based on the fundamental principles of analyticity and unitarity. In this framework, final-state interactions are fully taken into account. Our…

A recurrence equation is a discrete integrable equation whose solutions are all periodic and the period is fixed. We show that infinitely many recurrence equations can be derived from the information about invariant varieties of periodic…

We use the fact that certain cosets of the stabilizer of points are pairwise conjugate in a symmetric group $S_n$ in order to construct recurrence relations for enumerating certain subsets of $S_n$. Occasionally one can find `closed form'…

In this paper, we consider rational hypergeometric series of the form \[\frac{p}{\pi}= \sum_{k=0}^\infty u_k\quad\text{with}\quad u_k=\frac{\left(\frac{1}{2}\right)_k \left(q\right)_k \left(1-q\right)_k}{(k!)^3}(r+s\,k)\,t^k,\] where…

We introduce a series of discrete mappings, which is considered to be an extension of the Hietarinta-Viallet mapping with one parameter. We obtain the algebraic entropy for this mapping by obtaining the recurrence relation for the degrees…

For a fixed integer N, and fixed numbers b_1,...,b_N, we consider sequences, the nth term (a_n) of which is the sum of the squares of the terms in the expansion of (b_1 + ... + b_N)^n. In the case all b_i=1, we give a formula for a…

We present a natural, combinatorial problem whose solution is given by the meta-Fibonacci recurrence relation $a(n) = \sum_{i=1}^p a(n-i+1 - a(n-i))$, where $p$ is prime. This combinatorial problem is less general than those given in [3]…

In this work, we consider the properties of the two-term Machin-like formula and develop an algorithm for computing digits of $\pi$ by using its rational approximation. In this approximation, both terms are constructed by using a…

This paper presents different recursive formulas for computing the marginals and the normalizing constant of a Gibbs distribution $\pi$: The common thread is the use of the underlying Markov properties of such processes. The procedures are…

Given a linear recurrence of the form $c_n=a_1c_{n-1}+\cdots+a_j c_{n-j}$, it is well-known that $c_n=\sum_{r}p_r(n)r^n$, where the sum is taken over the set of characteristic roots and each $p_r(n)$ is some polynomial. We give a closed…

We provide a double-series formula for $\pi$ obtained using the Fourier series expansion of $1/\cos(x/4)$ and applying the Parseval-Plancherel identity. We show that such a formula involves the Grothendieck-Krivine constant, and that the…

The constant $\pi$ has fascinated scholars throughout the centuries, inspiring numerous formulas for its evaluation, such as infinite sums and continued fractions. Despite their individual significance, many of the underlying connections…

A consecutive pattern in a permutation $\pi$ is another permutation $\sigma$ determined by the relative order of a subsequence of contiguous entries of $\pi$. Traditional notions such as descents, runs and peaks can be viewed as particular…

For a fixed integer $k$, we define a sequence $A_k=(a_k(n))_{n\geq0}$ and a corresponding sparse subsequence $S_k$ using the cardinality of the $n$-th symmetric power of the set $\{1,2,\ldots, k\}$. For $k\in\{2,\dots,8\}$, we find…

In this note, using an idea from \cite{Amo-Carrillo-Sanchez} we derive some new series representations involving $\zeta(2n)$ and Euler numbers. Using a well-known series representation for the Clausen function, we also provide some new…

Two-term recurrence relations are supplied for indefinite integrals of functions that involve factors of the types ${P_2}^n$, ${P_3}^n$, ${P_4}^n$, ${P_1}^m {Q_1}^n$, $E_1 {P_1}^n$, ${P_1}^m {Q_2}^n$, $E_1 {P_2}^n$, ${P_2}^m {Q_2}^n$,…

We give asymptotic formulae for random matrix averages of derivatives of characteristic polynomials over the groups USp(2N), SO(2N) and O^-(2N). These averages are used to predict the asymptotic formulae for moments of derivatives of…

Let $\spt(n)$ be the number of smallest parts in the partitions of $n$. In this paper, we give some generalized Euler-like recursive formulas for the $\spt$ function in terms of Hecke trace of values of special twisted quadratic Dirichlet…

We derive a general recurrence relation for squares of Fibonacci-like numbers. Various properties are developed, including double binomial summation identites.

In our earlier publication we have shown how to compute by iteration a rational number ${u_{2,k}}$ in the two-term Machin-like formula for pi of kind…