The bilinear generating function for products of two Laguerre 2D polynomials with different arguments is calculated. It corresponds to the formula of Mehler for the generating function of products of two Hermite polyn...The bilinear generating function for products of two Laguerre 2D polynomials 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 operator disentanglement some operator identities are derived in an appendix. They allow calculating convolutions of Gaussian functions combined with polynomials in one- and two-dimensional case and are applied to evaluate the discussed generating functions.展开更多
This paper is confined to analyzing and implementing new spectral solutions of the fractional Riccati differential equation based on the application of the spectral tau method.A new explicit formula for approximating ...This paper is confined to analyzing and implementing new spectral solutions of the fractional Riccati differential equation based on the application of the spectral tau method.A new explicit formula for approximating the fractional derivatives of shifted Chebyshev polynomials of the second kind in terms of their original polynomials is established.This formula is expressed in terms of a certain terminating hypergeometric function of the type_(4)F_(3)(1).This hypergeometric function is reduced in case of the integer case into a certain terminating hypergeometric function of the type 3 F 2(1)which can be summed with the aid of Watson’s identity.Six illustrative examples are presented to ensure the applicability and accuracy of the proposed algorithm.展开更多
Explicit expressions of the Cotes numbers of the generalized Gaussian quadrature formulas for the Chebyshev nodes (of the first kind and the second kind) and their asymptotic behavior are given.
The purpose of this paper is to establish a formula of higher derivative by Faà di Bruno formula, and apply it to some known results to get some identities involving complete Bell polynomials.
The Chebyshev polynomials are harnessed as functions of the one parameter of the nondimensionalized differential equation for trinomial homogeneous linear differential equations of arbitrary order n that have constant...The Chebyshev polynomials are harnessed as functions of the one parameter of the nondimensionalized differential equation for trinomial homogeneous linear differential equations of arbitrary order n that have constant coefficients and exhibit vibration. The use of the Chebyshev polynomials allows calculation of the analytic solutions for arbitrary n in terms of the orthogonal Chebyshev polynomials to provide a more stable solution form and natural sensitivity analysis in terms of one parameter and the initial conditions in 6n + 7 arithmetic operations and one square root.展开更多
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 co...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.展开更多
文摘The bilinear generating function for products of two Laguerre 2D polynomials 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 operator disentanglement some operator identities are derived in an appendix. They allow calculating convolutions of Gaussian functions combined with polynomials in one- and two-dimensional case and are applied to evaluate the discussed generating functions.
文摘This paper is confined to analyzing and implementing new spectral solutions of the fractional Riccati differential equation based on the application of the spectral tau method.A new explicit formula for approximating the fractional derivatives of shifted Chebyshev polynomials of the second kind in terms of their original polynomials is established.This formula is expressed in terms of a certain terminating hypergeometric function of the type_(4)F_(3)(1).This hypergeometric function is reduced in case of the integer case into a certain terminating hypergeometric function of the type 3 F 2(1)which can be summed with the aid of Watson’s identity.Six illustrative examples are presented to ensure the applicability and accuracy of the proposed algorithm.
文摘Explicit expressions of the Cotes numbers of the generalized Gaussian quadrature formulas for the Chebyshev nodes (of the first kind and the second kind) and their asymptotic behavior are given.
基金Supported by the National Natural Science Foundation of China(Grant No.11601543,No.11601216,11701257)Supported by the NSF of Henan Province under Grant(No.172102410069)+1 种基金Supported by the NSF of Education Bureau of Henan Province under Grant(No.16B110009,18A110025)Supported by the Youth Foundation of Luoyang Normal university under Grant(No.2013-QNJJ-001)
文摘The purpose of this paper is to establish a formula of higher derivative by Faà di Bruno formula, and apply it to some known results to get some identities involving complete Bell polynomials.
文摘The Chebyshev polynomials are harnessed as functions of the one parameter of the nondimensionalized differential equation for trinomial homogeneous linear differential equations of arbitrary order n that have constant coefficients and exhibit vibration. The use of the Chebyshev polynomials allows calculation of the analytic solutions for arbitrary n in terms of the orthogonal Chebyshev polynomials to provide a more stable solution form and natural sensitivity analysis in terms of one parameter and the initial conditions in 6n + 7 arithmetic operations and one square root.
文摘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.
