Abstract:
We give one parameter generalizations of the Fibonacci and Lucas numbers denoted by $\{F_n(\th)\}$ and $\{L_n(\th)\}$, respectively. We evaluate the Hankel determinants with entries $\{1/F_{j+k+1}(\th): 0\le i,j \le n\}$ and $\{1/L_{j+k+1}(\th): 0\le i,j\le n\}$. We also find the entries in the inverse of $\{1/F_{j+k+1}(\th): 0\le i,j \le n\}$ and show that all its entries are integers. Some of the identities satisfied by the Fibonacci and Lucas numbers are extended to more general numbers. All integer solutions to three diophantine equtions related to the Pell equation are also found.

Abstract:
We prove some new results and unify the proofs of old ones involving complete monotonicity of expressions involving gamma and $q$-gamma functions, $0 < q < 1$. Each of these results implies the infinite divisibility of a related probability measure. In a few cases, we are able to get simple monotonicity without having complete monotonicity. All of the results lead to inequalities for these functions. Many of these were motivated by the bounds in a 1959 paper by Walter Gautschi. We show that some of the bounds can be extended to complex arguments.

Abstract:
The J-matrix method is extended to difference and q-difference operators and is applied to several explicit differential, difference, q-difference and second order Askey-Wilson type operators. The spectrum and the spectral measures are discussed in each case and the corresponding eigenfunction expansion is written down explicitly in most cases. In some cases we encounter new orthogonal polynomials with explicit three term recurrence relations where nothing is known about their explicit representations or orthogonality measures. Each model we analyze is a discrete quantum mechanical model in the sense of Odake and Sasaki [J. Phys. A: Math. Theor. 44 (2011), 353001, 47 pages].

Abstract:
A method is outlined to express a Tur n determinant of solutions of a three term recurrence relation as a weighted sum of squares. This method is shown to imply the positivity of Tur n determinants of symmetric Pollaczek polynomials, Lommel polynomials and q-Bessel functions.

Abstract:
We evaluate Fourier transform of a function with Hermite polynomials involved. An elementary proof is based on a combinatorial formula for Hermite polynomials.

Abstract:
The Askey-Wilson polynomials are orthogonal polynomials in$x = cos heta$, which are given as a terminating $_4phi_3$ basic hypergeometric series. The non-symmetric Askey-Wilson polynomials are Laurent polynomials in $z=e^{i heta}$, which are given as a sum of two terminating $_4phi_3$'s. They satisfy a biorthogonality relation. In this paper new orthogonality relations for single $_4phi_3$'s which are Laurent polynomials in~$z$ are given, which imply the non-symmetric Askey-Wilson biorthogonality. These results include discrete orthogonality relations. They can be considered as a classical analytic study of the results for non-symmetricAskey-Wilson polynomials which were previously obtained by affine Hecke algebra techniques.

Abstract:
We consider a class of rotationally invariant unitary random matrix ensembles where the eigenvalue density falls off as an inverse power law. Under a new scaling appropriate for such power law densities (different from the scaling required in Gaussian random matrix ensembles), we calculate exactly the two-level kernel that determines all eigenvalue correlations. We show that such ensembles belong to the class of critical ensembles.

Abstract:
We explore in this paper some orthogonal polynomials which are naturally associated to certain families of coherent states, often referred to as nonlinear coherent states in the quantum optics literature. Some examples turn out to be known orthogonal polynomials but in many cases we encounter a general class of new orthogonal polynomials for which we establish several qualitative results.

Abstract:
The q-difference analog of the classical ladder operators is derived for those orthogonal polynomials arising from a class of indeterminate moments problem.

Abstract:
We show connections between a special type of addition formulas and a theorem of Stieltjes and Rogers. We use different techniques to derive the desirable addition formulas. We apply our approach to derive special addition theorems for Bessel functions and confluent hypergeometric functions. We also derive several additions theorems for basic hypergeometric functions. Applications to the evaluation of Hankel determinants are also given .