The Experts below are selected from a list of 5385 Experts worldwide ranked by ideXlab platform
Yoon Seok Choun - One of the best experts on this subject based on the ideXlab platform.
-
The analytic solution for the power series expansion of Heun function
Annals of Physics, 2013Co-Authors: Yoon Seok ChounAbstract:Abstract The Heun function generalizes all well-known special functions such as Spheroidal Wave, Lame, Mathieu, and hypergeometric 2 F 1 , 1 F 1 and 0 F 1 functions. Heun functions are applicable to diverse areas such as theory of black holes, lattice systems in statistical mechanics, solution of the Schrodinger equation of quantum mechanics, and addition of three quantum spins. In this paper I will apply three term Recurrence Formula (Y.S. Choun, arXiv:1303.0806 , 2013) to the power series expansion in closed forms of Heun function (infinite series and polynomial) including all higher terms of A n ’s. Section 3 contains my analysis on applying the power series expansions of Heun function to a recent paper (R.S. Maier, Math. Comp. 33 (2007) 811–843). Due to space restriction final equations for the 192 Heun functions are not included in the paper, but feel free to contact me for the final solutions. Section 4 contains two additional examples using the power series expansions of Heun function. This paper is 3rd out of 10 in series “Special functions and three term Recurrence Formula (3TRF)”. See Section 5 for all the papers in the series. The previous paper in series deals with three term Recurrence Formula (3TRF). The next paper in the series describes the integral forms of Heun function and its asymptotic behaviors analytically.
-
special functions and reversible three term Recurrence Formula r3trf
arXiv: Classical Analysis and ODEs, 2013Co-Authors: Yoon Seok ChounAbstract:In the previous series "Special functions and three term Recurrence Formula (3TRF)", I generalize the three term Recurrence relation in the linear differential equation for the infinite series and polynomial which makes B_n term terminated including all higher terms of A_n's. In this series I will show how to obtain the Formula for the polynomial which makes A_n term terminated including all higher terms of B_n's and infinite series of its power series expansion. In the future series I will show you for the polynomial which makes A_n and B_n terms terminated at same time; the power series, integral formalism and generating function such as Heun, Mathieu, Lame and GCH equations will be constructed analytically. In chapter 1, I will generalize the three term Recurrence relation in linear differential equation in a backward for the infinite series and polynomial which makes A_n term terminated including all higher terms of B_n's. In chapters 2-9, I will apply reversible three term Recurrence Formula to (1) the power series expansion in closed forms, (2) its integral representation and (3) generating functions of Heun, Confluent Heun, GCH, Lame and Mathieu equations that consist of three term recursion relation for the infinite series and polynomial which makes A_n term terminated.
-
analytic solution for grand confluent hypergeometric function
arXiv: Mathematical Physics, 2013Co-Authors: Yoon Seok ChounAbstract:In previous paper I construct an approximative solution of the power series expansion in closed forms of Grand Confluent Hypergeometric (GCH) function only up to one term of A_n's [4]. And I obtain normalized constant and orthogonal relation of GCH function. In this paper I will apply three term Recurrence Formula [3] to the power series expansion in closed forms of GCH function (infinite series and polynomial) including all higher terms of A_n's. In general most of well-known special function with two recursive coefficients only has one eigenvalue for the polynomial case. However this new function with three recursive coefficients has infinite eigenvalues that make B_n's term terminated at specific value of index n because of three term Recurrence Formula [3]. This paper is 9th out of 10 in series "Special functions and three term Recurrence Formula (3TRF)". See section 6 for all the papers in the series. Previous paper in series deals with generating functions of Lame polynomial in the Weierstrass's form [28]. The next paper in the series describes the integral formalism and the generating function of GCH function [30].
-
the integral formalism and the generating function of grand confluent hypergeometric function
arXiv: Mathematical Physics, 2013Co-Authors: Yoon Seok ChounAbstract:Biconfluent Heun (BCH) function, a confluent form of Heun function, is the special case of Grand Confluent Hypergeometric (GCH) function: this has a regular singularity at x=0, and an irregular singularity at infinity of rank 2. In this paper I apply three term Recurrence Formula (3TRF) [arXiv:1303.0806] to the integral formalism of GCH function including all higher terms of A_n's and the generating function for the GCH polynomial which makes B_n term terminated. I show how to transform power series expansion in closed forms of GCH equation to its integral representation analytically. This paper is 10th out of 10 in series "Special functions and three term Recurrence Formula (3TRF)". See section 6 for all the papers in the series. The previous paper in the series describes the power series expansion in closed forms of GCH equation and its asymtotic behaviours. [arXiv:1303.0813]
-
generalization of the three term Recurrence Formula and its applications
arXiv: Mathematical Physics, 2013Co-Authors: Yoon Seok ChounAbstract:The history of linear differential equations is over 350 years. By using Frobenius method and putting the power series expansion into linear differential equations, the recursive relation of coefficients starts to appear. There can be between two and infinity number of coefficients in the Recurrence relation in the power series expansion. During this period mathematicians developed analytic solutions of only two term recursion relation in closed forms. Currently the analytic solution of three term Recurrence relation is unknown. In this paper I will generalize the three term Recurrence relation in the linear differential equation. This paper is 2nd out of 10 in series "Special functions and three term Recurrence Formula (3TRF)". The next paper in series deals with the power series expansion in closed forms of Heun function by Choun [arXiv:1303.0830]. The rest of the papers in the series show how to solve mathematical equations having three term recursion relations and go on producing the exact solutions of some of the well known special functions including: Mathieu, Heun, Biconfluent Heun and Lame equations. See section IX for all the papers and short descriptions in the series.
Kohei Watabe - One of the best experts on this subject based on the ideXlab platform.
-
analysis of the convergence speed of the arimoto blahut algorithm by the second order Recurrence Formula
IEEE Transactions on Information Theory, 2021Co-Authors: Kenji Nakagawa, Yoshinori Takei, Shin-ichiro Hara, Kohei WatabeAbstract:In this paper, we investigate the convergence speed of the Arimoto-Blahut algorithm. For many channel matrices, the convergence speed is exponential, but for some channel matrices it is slower than exponential. By analyzing the Taylor expansion of the defining function of the Arimoto-Blahut algorithm, we will make the conditions clear for the exponential or slower convergence. The analysis of the slow convergence in this paper is new. Based on this analysis, we will compare the convergence speeds of the Arimoto-Blahut algorithm numerically with the values obtained in our theorems for several channel matrices. The purpose of this paper is to obtain a complete understanding of the convergence speed of the Arimoto-Blahut algorithm.
-
Analysis of the Convergence Speed of the Arimoto-Blahut Algorithm by the Second Order Recurrence Formula.
arXiv: Information Theory, 2020Co-Authors: Kenji Nakagawa, Yoshinori Takei, Shin-ichiro Hara, Kohei WatabeAbstract:In this paper, we investigate the convergence speed of the Arimoto-Blahut algorithm. For many channel matrices the convergence is exponential, but for some channel matrices it is slower than exponential. By analyzing the Taylor expansion of the defining function of the Arimoto-Blahut algorithm, we will make the conditions clear for the exponential or slower convergence. The analysis of the slow convergence is new in this paper. Based on the analysis, we will compare the convergence speed of the Arimoto-Blahut algorithm numerically with the values obtained in our theorems for several channel matrices. The purpose of this paper is a complete understanding of the convergence speed of the Arimoto-Blahut algorithm.
Kenji Nakagawa - One of the best experts on this subject based on the ideXlab platform.
-
analysis of the convergence speed of the arimoto blahut algorithm by the second order Recurrence Formula
IEEE Transactions on Information Theory, 2021Co-Authors: Kenji Nakagawa, Yoshinori Takei, Shin-ichiro Hara, Kohei WatabeAbstract:In this paper, we investigate the convergence speed of the Arimoto-Blahut algorithm. For many channel matrices, the convergence speed is exponential, but for some channel matrices it is slower than exponential. By analyzing the Taylor expansion of the defining function of the Arimoto-Blahut algorithm, we will make the conditions clear for the exponential or slower convergence. The analysis of the slow convergence in this paper is new. Based on this analysis, we will compare the convergence speeds of the Arimoto-Blahut algorithm numerically with the values obtained in our theorems for several channel matrices. The purpose of this paper is to obtain a complete understanding of the convergence speed of the Arimoto-Blahut algorithm.
-
Analysis of the Convergence Speed of the Arimoto-Blahut Algorithm by the Second Order Recurrence Formula.
arXiv: Information Theory, 2020Co-Authors: Kenji Nakagawa, Yoshinori Takei, Shin-ichiro Hara, Kohei WatabeAbstract:In this paper, we investigate the convergence speed of the Arimoto-Blahut algorithm. For many channel matrices the convergence is exponential, but for some channel matrices it is slower than exponential. By analyzing the Taylor expansion of the defining function of the Arimoto-Blahut algorithm, we will make the conditions clear for the exponential or slower convergence. The analysis of the slow convergence is new in this paper. Based on the analysis, we will compare the convergence speed of the Arimoto-Blahut algorithm numerically with the values obtained in our theorems for several channel matrices. The purpose of this paper is a complete understanding of the convergence speed of the Arimoto-Blahut algorithm.
G A Ghazal - One of the best experts on this subject based on the ideXlab platform.
-
Recurrence Formula for expectations of products of bilinear forms and expectations of bilinear forms and random matrices
Statistics & Probability Letters, 2000Co-Authors: G A GhazalAbstract:Abstract In this paper, we derive a Recurrence Formula for evaluating mathematical expectations of ∏ j=1 n (X 2j−1 ′A j X 2j )=X 1 ′A 1 X 2 .X 3 ′A 2 X 4 …X 2n−1 ′A n X 2n , where X j for j =1,2,…,2 n is p ×1 random vectors that are jointly normally distributed with E(X i )=0, i=1,2,…,2n, E(X i X j ′)=φ ij , i,j=1,2,…,2n and A j , j=1,2,…,n are p × p nonstochastic matrices. Subsequently, this Recurrence Formula is used to derive the expectations of products of bilinear forms and random matrices, namely, E(X i ′A 1 X j .X r X s ′) and E(X i ′A 1 X j .X k ′A 2 X l .X r X s ′). Further, some related results will also be reported.
-
Recurrence Formula for expectations of products of quadratic forms
Statistics & Probability Letters, 1996Co-Authors: G A GhazalAbstract:Abstract In this paper, we derive a Recurrence Formula for evaluating mathematical expectations of X′A 1 XX′A 2 X…X′A n X where X ∼ Np(0, Φ) and Aj, j = 1, 2, …, n are (p × p) nonstochastic symmetric matrices. Subsequently, this Recurrence Formula is applied to some specific problems. Various related results will also be reported.
Shin-ichiro Hara - One of the best experts on this subject based on the ideXlab platform.
-
analysis of the convergence speed of the arimoto blahut algorithm by the second order Recurrence Formula
IEEE Transactions on Information Theory, 2021Co-Authors: Kenji Nakagawa, Yoshinori Takei, Shin-ichiro Hara, Kohei WatabeAbstract:In this paper, we investigate the convergence speed of the Arimoto-Blahut algorithm. For many channel matrices, the convergence speed is exponential, but for some channel matrices it is slower than exponential. By analyzing the Taylor expansion of the defining function of the Arimoto-Blahut algorithm, we will make the conditions clear for the exponential or slower convergence. The analysis of the slow convergence in this paper is new. Based on this analysis, we will compare the convergence speeds of the Arimoto-Blahut algorithm numerically with the values obtained in our theorems for several channel matrices. The purpose of this paper is to obtain a complete understanding of the convergence speed of the Arimoto-Blahut algorithm.
-
Analysis of the Convergence Speed of the Arimoto-Blahut Algorithm by the Second Order Recurrence Formula.
arXiv: Information Theory, 2020Co-Authors: Kenji Nakagawa, Yoshinori Takei, Shin-ichiro Hara, Kohei WatabeAbstract:In this paper, we investigate the convergence speed of the Arimoto-Blahut algorithm. For many channel matrices the convergence is exponential, but for some channel matrices it is slower than exponential. By analyzing the Taylor expansion of the defining function of the Arimoto-Blahut algorithm, we will make the conditions clear for the exponential or slower convergence. The analysis of the slow convergence is new in this paper. Based on the analysis, we will compare the convergence speed of the Arimoto-Blahut algorithm numerically with the values obtained in our theorems for several channel matrices. The purpose of this paper is a complete understanding of the convergence speed of the Arimoto-Blahut algorithm.
