Convexity properties are often useful in characterizing and nding bounds for special function and their zeros, as well as in questions concerning the existence and uniqueness of zeros in certain intervals. In this survey paper, we describe some work related to the gamma function, the q-gamma function, Bessel and cylinder functions and the Hermite function.

We consider the approximation by a spectral method of the solution of the Cauchy problem for a scalar conservation law in one dimension posed in the whole real line. We analyze a spectral viscosity method in which the orthogonal basis considered is the one of Hermite functions. We prove the convergence of the approximate solution to the unique entropy solution of the problem by using compensate...

A two-dimensional generalized oscillator with time-dependent parameters is considered to study the two-mode squeezing phenomena. Specific choices of the parameters are used to determine the dispersion matrix and analytic expressions, in terms of standard hermite polynomials, of the wavefunctions and photon distributions.

A new kind of deformed calculus was introduced recently in studying of parabosonic coordinate representation. Based on this deformed calculus, a new deformation of Hermite polynomials is proposed, its some properties such as generating function, orthonormality, differential and integral representaions, and recursion relations are also discussed in this paper. As its applications, we calculate e...

Consider the space L2 C, dμ z , where dμ z e−|z| 2 dz ∧ dz is the Gaussian measure, and its generalized Bargmann subspaces Em which are the null kernels of the operator Δ −∂2/∂z∂z z ∂/∂z − mI; m 0, 1, . . . . In this work, we present an other construction of Em following the Hermite functions which allows us to define a family of generalized Bargmann transform Bm which maps isometrically Em int...

We make precise some properties of the Hermite function in relation with Morse theory introduced by Avner Ash in his papers [Ash1] and [Ash2] and with the cellular decomposition of the space of positive definite quadratic forms. We also establish a link between Ash’s and Bavard’s mass formulae.

Carl de Boor 8oct02 Abstract The Leibniz formula, for the divided difference of a product, and Opitz’s formula, for the divided difference table of a function as the result of evaluating that function at a certain matrix, are shown to be special cases of a formula available for the coefficients, with respect to any basis, of an ‘ideal’ or ‘Hermite’ polynomial interpolant, in any number of varia...

Let μ be a probability measure on the real line with finite moments of all orders. Apply the Gram-Schmidt orthogonalization process to the system {1, x, x, . . . , xn, . . . } to get orthogonal polynomials Pn(x), n ≥ 0, which have leading coefficient 1 and satisfy (x − αn)Pn(x) = Pn+1(x) + ωnPn−1(x). In general it is almost impossible to use this process to compute the explicit form of these po...

We consider the set of Stieltjes moment sequences, for which every positive power is again a Stieltjes moment sequence, we and prove an integral representation of the logarithm of the moment sequence in analogy to the Lévy-Khintchine representation. We use the result to construct product convolution semigroups with moments of all orders and to calculate their Mellin transforms. As an applicatio...

We compute the integrals of products of Hermite functions using the generating functions. The precise asymptotics of products of 4 Hermite functions are presented below. This estimate is relevant for the corresponding cubic nonlinear equation.