On some identities for the Fibonomial coefficients via generating function

Some new identities for the Fibonomial coefficients are derived. These identities are related to the generating function of the kth powers of the Fibonacci numbers. Proofs are based on manipulation with the generating function of the sequence of “signed Fibonomial triangle”.     

Quadratic sums of Gaussian q-binomial coefficients and Fibonomial coefficients

The Ramanujan Journal - In this paper we evaluate quadratic sums of Gaussian q-binomial coefficients with two additional parameters. We obtain a general summation theorem using a combination of...     

On some identities of H-function

In this paper we establish some identities of H-functions involving complex arguments.     

一些包含Lucas数的恒等式

利用第一类Chebyshev多项式的性质以及其与Lucas数的关系得到了关于Lucas数立方的一些恒等式.     

On some determinantal identities formation laws

We show that Muir’s law of extensible minors, Cayley’s law of complementaries and Jacobi’s identity for minors of the adjugate [Determinantal identities Linear Algebra and its Applications 52/53 (1983) pp. 769–791] are equivalent. We also show our generalization of Mühlbach/Muir’s extension principle [A generalization of Mühlbach’s extension principle for determinantal identities. Linear and Multilinear Algebra 61 (10) (2013) pp. 1363–1376] is equivalent to its previous form derived by Mühlbach. As a corollary, we show that Mühlbach–Gasca–(Lopez-Carmona)–Ramirez identity [A generalization of Sylvester’s identity on determinants and some applications. Linear Algebra and its Applications 66 (1985) pp. 221–234/On extending determinantal identities. Linear Algebra and its Applications 132 (1990) pp. 145–162] is equivalent to its generalization found by Beckermann and Mühlbach [A general determinantal identity of Sylvester type and some applications. Linear Algebra and its Applications 197,198 (1994) pp. 93–112].     

On certain identities for ratios of theta-functions and some new modular equations of mixed degree

In this paper, we derive certain identities for ratios of theta-functions. As applications of the identities, we establish certain new modular equations of mixed degree in the theory of signature 3, which are analogous to Ramanujan-Weber type modular equations and Ramanujan-Schläfli type mixed modular equations.     

On some identities and inequalities for frames in Hilbert spaces

We give new Parseval type identities and inequalities for frames in Hilbert spaces. Our results generalize the remarkable results obtained recently by R. Balan, P.G. Casazza, D. Edidin, and G. Kutyniok.     

M-bonomial coefficients and their identities

In this note, we introduce M-bonomial coefficients or (M-bonacci binomial coefficients). These are similar to the binomial and the Fibonomial (or Fibonacci–binomial) coefficients and can be displayed in a triangle similar to Pascal's triangle from which some identities become obvious.     

On a Congruence Involving Generalized Fibonomial Coefficients

Let (F_n)_n≥0 be the Fibonacci sequence. For 1 ≤ k ≤ m, the Fibonomial coefficient is defined as

$${\left[ {\begin{array}{*{20}{c}} n \\ k \end{array}} \right]_F} = \frac{{{F_{n - k + 1}} \cdots {F_{n - 1}}{F_n}}}{{{F_1} \cdots {F_k}}}$$

. In 2013, Marques, Sellers and Trojovský proved that if p is a prime number such that p ≡ ±1 (mod 5), then p?\({\left[ {\begin{array}{*{20}{c}} {{p^{a + 1}}} \\ {{p^a}} \end{array}} \right]_F}\) for all integers a ≥ 1. In 2010, in particular, Kilic generalized the Fibonomial coefficients for

$${\left[ {\begin{array}{*{20}{c}} n \\ k \end{array}} \right]_{F,m}} = \frac{{{F_{\left( {n - k + 1} \right)m}} \cdots {F_{\left( {n - 1} \right)m}}{F_{nm}}}}{{{F_m} \cdots {F_{km}}}}$$

. In this note, we generalize Marques, Sellers and Trojovský result to prove, in particular, that if p ≡ ±1 (mod 5), then \({\left[ {\begin{array}{*{20}{c}} {{p^{a + 1}}} \\ {{p^a}} \end{array}} \right]_{F,m}} \equiv 1\) (mod p), for all a ≥ 0 and m ≥ 1.

     

On trace identities and universal eigenvalue estimates for some partial differential operators

In this article, we prove and exploit a trace identity for the spectra of Schrödinger operators and similar operators. This identity leads to universal bounds on the spectra, which apply to low-lying eigenvalues, eigenvalue asymptotics, and to partition functions (traces of heat operators). In many cases they are sharp in the sense that there are specific examples for which the inequalities are saturated. Special cases corresponding to known inequalities include those of Hile and Protter and of Yang.

     

On the Laplacian coefficients of graphs under some transformations

We generalize three approaches on graph transformations, respectively, from Stevanovi? and Ili? (2009) [16] and Tan (2011) [19]. We also generalize an approach of graph transformations on the spectral radius of adjacency matrix into the Laplacian coefficients of graphs from Li and Feng (1979) [26]. Moreover, we determine the unique tree having the third maximal Laplacian coefficients among all n-vertex trees.     

Jack polynomials and some identities for partitions

We prove an identity about partitions involving new combinatorial coefficients. The proof given is using a generating function. As an application we obtain the explicit expression of two shifted symmetric functions, related with Jack polynomials. These quantities are the moments of the ``-content' random variable with respect to some transition probability distributions.

     

On some generalizations of Ramanujan’s continued fraction identities

In this note we establish continued fraction developments for the ratios of the basic hypergeometric function₂ϕ₁(a,b;c;x) with several of its contiguous functions. We thus generalize and give a unified approach to establishing several continued fraction identities including those of Srinivasa Ramanujan.     

Positivity for some Satake coefficients

Let G be a general linear group over a local field F. We consider the matrix describing the Satake isomorphism with respect to natural bases. We give a simple proof for the positivity of all matrix coefficients that are not obviously zero. The arguments are elementary and more direct than Rapoport's original proof. Mathematics Subject Classification (2000):22E35     

关于几类图族伴随多项式的第四项系数

主要研究了几类图族伴随多项式第四项系数的规律，此结果有助于进一步讨论这些图族补图的色唯一性、色等价划分．