Mehler Formula Research Articles

Generating Functions for Products of Special Laguerre 2D and Hermite 2D Polynomials

The bilinear generating function for products of two Laguerre 2D polynomials Lm;n(z; z0) with different arguments is calculated. It corresponds to the formula of Mehler for the generating function of products of two Hermite polynomials. Furthermore, the generating function for mixed products of Laguerre 2D and Hermite 2D polynomials and for products of two Hermite 2D polynomials is calculated. A set of infinite sums over products of two Laguerre 2D polynomials as intermediate step to the generating function for products of Laguerre 2D polynomials is evaluated but these sums possess also proper importance for calculations with Laguerre polynomials. With the technique of SU(1; 1) operator disentanglement some operator identities are derived in an appendix. They allow to calculate convolutions of Gaussian functions combined with polynomials in one- and two-dimensional case and are applied to evaluate the discussed generating functions. Keywords: Laguerre and Hermite polynomials, Laguerre 2D polynomials, Jacobi polynomials, Mehler formula, SU(1; 1) operator disentanglement, Gaussian convolutions.

A Multinomial Theorem for Hermite Polynomials and Financial Applications

Different aspects of mathematical finance benefit from the use Hermite polynomials, and this is particularly the case where risk drivers have a Gaussian distribution. They support quick analytical methods which are computationally less cumbersome than a full-fledged Monte Carlo framework, both for pricing and risk management purposes. In this paper, we review key properties of Hermite polynomials before moving on to a multinomial expansion formula for Hermite polynomials, which is proved using basic methods and corrects a formulation that appeared before in the financial literature. We then use it to give a trivial proof of the Mehler formula. Finally, we apply it to no arbitrage pricing in a multi-factor model and determine the empirical futures price law of any linear combination of the underlying factors.

Probabilistic proofs of some q-identities

In this paper, we use probabilistic method to prove the three well-known q-identities: the identity of Andrews, Mehler's formula, the Jackson q-analogue of the Euler transform.

The q-Exponential Operator and Generalized Rogers-Szegö Polynomials

This paper is mainly concerned with using -exponential operator in proving the identities that involve the generalized Rogers-Szego polynomials . We introduce some new roles of the -exponential operator and prove that the generalized Rogers-Szego polynomials can be represented by the -exponential operator, so we use this operator and it’s roles in proving the basic identities of given in [7, 8] which are: generating function, Mehler’s formula and Rogers formula. Then we introduce several extensions of identities such that: the extended generating function, extended Mehler’s formula, extended Rogers formula and another extended identities. These extended identities of the generalized Rogers-Szego polynomials can be considered a general form of the corresponding identities for the classical Rogers-Szego polynomials when b=1.

Uniqueness Result in the Cauchy Dirichlet Problem via Mehler Kernel

Using the Mehler kernel, a uniqueness theorem in the Cauchy Dirichlet problem for the Hermite heat equation with homogeneous Dirichlet boundary conditions on a class P of bounded functions U(x, t) with certain growth on Ux(x, t) is established.

A PROBABILISTIC VERSION OF MEHLER'S FORMULA

In this paper, we use Lebesgue's dominated convergence theorem to give a probabilistic version of Mehler's formula for the well-known Rogers-Szegö polynomials. Applications of the probabilistic version are also given.

A characterization of distributions of exponential growth with support in a regular closed set

In this paper we shall give the representation theorem for the space of distributions of exponential growth supported by a regular closed set, and characterize this space by the heat kernel method. Furthermore we shall see that we can also characterize this space by using the Mehler kernel.

The operator G(a; b;Dq) for the polynomials Wn(x; y; a; b; q)

We give an identity which can be regarded as a basic result for this paper. Inspiredby this identity we introduce an operator G(a, b;Dq). The exponential operator R(bDq)defined by Saad and Sukhi [11] can be considered as a special case of the operator G(a, b;Dq)for a = 0. Also we introduce a polynomials Wn(x, y, a, b; q). Al-Salam-Carlitz polynomialsUn(x, y, b; q) [4] is a special case of Wn(x, y, a, b; q) for a = 0. So all the identities for thepolynomials Wn(x, y, a, b; q) are extensions of formulas for the Al-Salam-Carlitz polynomialsUn(x, y, a; q). We give an operator proof for the generating function, the Rogers formula andthe Mehlers formula for Wn(x, y, a, b; q). Rogers formula leads to the inverse linearizationformula. We give another Rogers-type formula for the polynomials Wn(x, y, a, b; q).

An estimate for bounded solutions of the Hermite heat equation

An estimate result on the partial derivatives of the Mehler kernel E(x; ;t ) for t > 0 is rst established. Particularly for 0 < t < 1, it extends the estimate result given by S. Thangavelu in his monograph A lecture notes on Hermite and Laguerre expansions on the order of the partial derivative of the Mehler kernel with respect to the space variable. Furthermore, for each m 2 N0, a growth estimate on the partial derivative @ m U(x;t) @xm of all bounded solutions U(x;t) of the Cauchy Dirichlet problem for the Hermite heat equation is established.

Mehler formulae for matching polynomials of graphs and independence polynomials of clawfree graphs

The independence polynomial of a graph G is the polynomial ∑Ix|I|, summed over all independent subsets I⊆V(G). We show that if G is clawfree, then there exists a Mehler formula for its independence polynomial. This was proved for matching polynomials in Lass (2004) [19] and extends the combinatorial proof of the Mehler formula found by Foata (1978) [9]. It implies immediately that all the roots of the independence polynomial of a clawfree graph are real, answering a question posed by Hamidoune (1990) [14] and Stanley (1998) [28] and solved by Chudnovsky and Seymour (2007) [6]. We also prove a Mehler formula for the multivariate matching polynomial, extending results of Lass (2004) [19].

Towards a q-analogue of the Kibble–Slepian formula in 3 dimensions

We study a generalization of the Kibble–Slepian (KS) expansion formula in 3 dimensions. The generalization is obtained by replacing the Hermite polynomials by the q-Hermite ones. If such a replacement would lead to non-negativity for all allowed values of parameters and for all values of variables ranging over certain Cartesian product of compact intervals then we would deal with a generalization of the 3-dimensional Normal distribution. We show that this is not the case. Nevertheless we indicate other applications of so-generalized KS formula. Namely we use it to sum certain kernels built of the Al-Salam–Chihara polynomials for the cases that were not considered by other authors. One of such kernels sums up to the Askey–Wilson density disclosing its new, interesting properties. In particular we are able to obtain a generalization of the 2-dimensional Poisson–Mehler formula. We also pose several open questions.

Orthogonality of Hermite polynomials in superspace and Mehler type formulae

In this paper, Hermite polynomials related to quantum systems with orthogonal O(m)-symmetry, finite reflection group symmetry 𝒢<O(m), symplectic symmetry Sp(2n) and superspace symmetry O(m)×Sp(2n) are considered. After an overview of the results for O(m) and 𝒢, the orthogonality of the Hermite polynomials related to Sp(2n) is obtained with respect to the Berezin integral. As a consequence, an extension of the Mehler formula for the classical Hermite polynomials to Grassmann algebras is proved. Next, Hermite polynomials in a full superspace with O(m)×Sp(2n)-symmetry are considered. It is shown that they are not orthogonal with respect to the canonically defined inner product. However, a new inner product is introduced, which behaves correctly with respect to the structure of harmonic polynomials on superspace. This inner product allows to restore the orthogonality of the Hermite polynomials and also restores the hermiticity of a class of Schrödinger operators in superspace. Subsequently, a Mehler formula for the full superspace is obtained, thus yielding an eigenfunction decomposition of the super Fourier transform. Finally, the new results for the Sp(2n)- and O(m)×Sp(2n)-symmetry are compared with the results in the different types of symmetry.

Expansions of one density via polynomials orthogonal with respect to the other

We expand the Chebyshev polynomials and some of its linear combination in linear combinations of the q-Hermite, the Rogers ( q-utraspherical) and the Al-Salam–Chihara polynomials and vice versa. We use these expansions to obtain expansions of some densities, including q-Normal and some related to it, in infinite series constructed of the products of the other density times polynomials orthogonal to it, allowing deeper analysis and discovering new properties. On the way we find an easy proof of expansion of the Poisson–Mehler kernel as well as its reciprocal. We also formulate simple rule relating one set of orthogonal polynomials to the other given the properties of the ratio of the respective densities of measures orthogonalizing these polynomials sets.

Topological Graph Polynomial and Quantum Field Theory Part II: Mehler Kernel Theories

We define a new topological polynomial extending the Bollobás–Riordan one, which obeys a four-term reduction relation of the deletion/contraction type and has a natural behaviour under partial duality. This allows to write down a completely explicit combinatorial evaluation of the polynomials, occurring in the parametric representation of the non-commutative Grosse–Wulkenhaar quantum field theory. An explicit solution of the parametric representation for commutative field theories based on the Mehler kernel is also provided.

Sociopolitical Risk Management Through Prediction Markets

We show that market prices can reveal information about the sociology of the agent population. This is done in the setting of prediction markets where the contract pay-off depends on a future event. We find that the development of market prices encodes variables such as the social climate and the interaction between agents. This shows that prediction markets can be used as a tool for the management of sociopolitical risks in large groups. The paper uses a well-known sociodynamic model for the formation of the collective opinion and derives a formula for the price of prediction market contracts. The stochastic process for the opinion formation is approximated by an Ito process through van Kampen's size expansion technique. The contract prices arise as solutions of the Euclidean Schroedinger equation via Mehler's formula. Our work also provides a theoretical framework justifying assumptions generally used for describing prediction market prices in a continuous-time setting.

An existence result of the Cauchy Dirichlet problem for the Hermite heat equation

Using the Mehler kernel, we give an existence result of the Cauchy Dirichlet problem for the Hermite heat equation with homogeneous Dirichlet boundary conditions and continuous and bounded Cauchy data vanishing at x = 0.

A Unified Distortion Analysis of Nonlinear Power Amplifiers with Memory Effects for OFDM Signals

Nonlinear distortions in power amplifiers (PAs) generate spectral regrowth at the output, which causes interference to adjacent channels and errors in digitally modulated signals. This paper presents a novel method to evaluate adjacent channel leakage power ratio (ACPR) and error vector magnitude (EVM) from the amplitude-to-amplitude (AM/AM) and amplitude-to-phase (AM/PM) characteristics. The transmitted signal is considered to be complex Gaussian distributed in orthogonal frequency-division multiplexing (OFDM) systems. We use the Mehler formula to derive closed-form expressions of the PAs output power spectral density (PSD), ACPR and EVM for memoryless PA and memory PA respectively. We inspect the derived relationships using an OFDM signal in the IEEE 802.11a WLAN standard. Simulation results show that the proposed method is appropriate to predict the ACPR and EVM values of the nonlinear PA output in OFDM systems, when the AM/AM and AM/PM characteristics are known.

On Wiener filtering of certain locally stationary stochastic processes

We study linear minimum mean squared error filters for continuous-time second-order stochastic processes that are locally stationary in Silverman's sense. We show that the optimal filter is rarely locally stationary even when the covariance functions have Gaussian shape. Using Mehler's formula we derive series expansions of the filter kernel for locally stationary covariances that are determined by Gaussians.

Kalman filter and quantization

We give an interpretation of the problem of filtering of diffusion processes as a quantization problem. Based on this, we show that the classical Kalman-Bucy linear filter describes a flow of automorphisms of the Heisenberg algebra. We obtain new formulas for the unnormalized conditional density in the linear case, a new interpretation of the Mehler formula for the fundamental solution of the Schrodinger operator for a harmonic oscillator, and formulas for a regularized determinant of a Sturm-Liouville operator.

Operator identities involving the bivariate Rogers–Szegö polynomials and their applications to the multiple q-series identities

In this paper, we first give several operator identities involving the bivariate Rogers–Szegö polynomials. By applying the technique of parameter augmentation to the multiple q-binomial theorems given by Milne [S.C. Milne, Balanced ϕ 2 3 summation theorems for U ( n ) basic hypergeometric series, Adv. Math. 131 (1997) 93–187], we obtain several new multiple q-series identities involving the bivariate Rogers–Szegö polynomials. These include multiple extensions of Mehler's formula and Rogers's formula. Our U ( n + 1 ) generalizations are quite natural as they are also a direct and immediate consequence of their (often classical) known one-variable cases and Milne's fundamental theorem for A n or U ( n + 1 ) basic hypergeometric series in Theorem 1.49 of [S.C. Milne, An elementary proof of the Macdonald identities for A l ( 1 ) , Adv. Math. 57 (1985) 34–70], as rewritten in Lemma 7.3 on p. 163 of [S.C. Milne, Balanced ϕ 2 3 summation theorems for U ( n ) basic hypergeometric series, Adv. Math. 131 (1997) 93–187] or Corollary 4.4 on pp. 768–769 of [S.C. Milne, M. Schlosser, A new A n extension of Ramanujan's ψ 1 1 summation with applications to multilateral A n series, Rocky Mountain J. Math. 32 (2002) 759–792].