scholarly journals The derivative connecting problems for some classical polynomials

2019 ◽  
Vol 11 (2) ◽  
pp. 431-441
Author(s):  
A. Ramskyi ◽  
N. Samaruk ◽  
O. Poplavska
Keyword(s):  
Closed Form ◽  
Quantum Chemical ◽  
Applied Mathematics ◽  
Mathematical Physics ◽  
Closed Form Expression ◽  
Form Expression ◽  
Connection Problem ◽  
Classical Polynomials ◽  
Different Types ◽  
General Connection

Given two polynomial sets $\{ P_n(x) \}_{n\geq 0},$ and $\{ Q_n(x) \}_{n\geq 0}$ such that $$\deg ( P_n(x) ) =n, \deg ( Q_n(x) )=n.$$ The so-called the connecting problem between them asks to find the coefficients $\alpha_{n,k}$ in the expression $\displaystyle Q_n(x) =\sum_{k=0}^{n} \alpha_{n,k} P_k(x).$ Let $\{ S_n(x) \}_{n\geq 0}$ be another polynomial set with $\deg ( S_n(x) )=n.$ The general connection problem between them consists in finding the coefficients $\alpha^{(n)}_{i,j}$ in the expansion $$Q_n(x) =\sum_{i,j=0}^{n} \alpha^{(n)}_{i,j} P_i(x) S_{j}(x).$$ The connection problem for different types of polynomials has a long history, and it is still of interest. The connection coefficients play an important role in many problems in pure and applied mathematics, especially in combinatorics, mathematical physics and quantum chemical applications. For the particular case $Q_n(x)=P'_{n+1}(x)$ the connection problem is called the derivative connecting problem and the general derivative connecting problem associated to $\{ P_n(x) \}_{n\geq 0}.$ In this paper, we give a closed-form expression of the derivative connecting problems for well-known systems of polynomials.

scholarly journals The inverse and derivative connecting problems for some hypergeometric polynomials

2018 ◽  
Vol 10 (2) ◽  
pp. 235-247
Author(s):  
L. Bedratyuk ◽  
A. Bedratuyk
Keyword(s):  
Applied Mathematics ◽  
Closed Form Expression ◽  
Gauss Hypergeometric Function ◽  
Real Numbers ◽  
Form Expression ◽  
Pochhammer Symbol ◽  
Hypergeometric Polynomials ◽  
Connection Problem ◽  
Different Types ◽  
Polynomial Family

Given two  polynomial sets $\{ P_n(x) \}_{n\geq 0},$ and $\{ Q_n(x) \}_{n\geq 0}$  such that $\deg ( P_n(x) ) = \deg ( Q_n(x) )=n.$ The so-called connection problem between them asks to find coefficients  $\alpha_{n,k}$ in the expression $\displaystyle Q_n(x) =\sum_{k=0}^{n} \alpha_{n,k} P_k(x).$ The connection problem for  different types of polynomials has a long history, and it is still of interest. The connection coefficients play an important role in many problems in pure and applied mathematics, especially in combinatorics, mathematical physics and quantum chemical applications. For the particular case $Q_n(x)=x^n$  the connection problem  is called the inversion problem associated to $\{P_n(x)\}_{n\geq 0}.$ The particular case $Q_n(x)=P'_{n+1}(x)$ is called the derivative connecting problem for polynomial family $\{ P_n(x) \}_{n\geq 0}.$ In this paper, we give a closed-form expression of the inversion and the derivative coefficients for hypergeometric polynomials of the form $${}_2 F_1 \left[ \left. \begin{array}{c} -n, a \\ b \end{array} \right | z \right], {}_2 F_1 \left[ \left. \begin{array}{c} -n, n+a \\ b \end{array} \right | z \right], {}_2 F_1 \left[ \left. \begin{array}{c} -n, a \\ \pm n +b \end{array} \right | z \right],$$ where $\displaystyle {}_2 F_1 \left[ \left. \begin{array}{c} a, b \\ c \end{array} \right | z \right] =\sum_{k=0}^{\infty} \frac{(a)_k (b)_k}{(c)_k} \frac{z^k}{k!},$ is the Gauss hypergeometric function and $(x)_n$ denotes the Pochhammer symbol defined by $$\displaystyle  (x)_n=\begin{cases}1, n=0, \\x(x+1)(x+2)\cdots (x+n-1) , n>0.\end{cases}$$ All polynomials are considered over the field of real numbers.

scholarly journals Synthesizing of Planar and Conformal Double-Sided Parallel-Cross Dipoles FSS Based on Closed-Form Expression

IEEE Access ◽  
2021 ◽  
pp. 1-1
Author(s):  
Yassine Zouaoui ◽  
Larbi Talbi ◽  
Khelifa Hettak ◽  
Naresh K. Darimireddy
Keyword(s):  
Closed Form ◽  
Closed Form Expression ◽  
Form Expression

Distribution of Consensus in a Broadcasting-based Consensus-forming Algorithm

2021 ◽  
Vol 48 (3) ◽  
pp. 91-96
Author(s):  
Shigeo Shioda
Keyword(s):  
Distribution Function ◽  
Probability Density Function ◽  
Closed Form ◽  
Probability Density ◽  
Infinite Number ◽  
Stable Distribution ◽  
Random Variable ◽  
Cauchy Distribution ◽  
Closed Form Expression ◽  
Form Expression

The consensus achieved in the consensus-forming algorithm is not generally a constant but rather a random variable, even if the initial opinions are the same. In the present paper, we investigate the statistical properties of the consensus in a broadcasting-based consensus-forming algorithm. We focus on two extreme cases: consensus forming by two agents and consensus forming by an infinite number of agents. In the two-agent case, we derive several properties of the distribution function of the consensus. In the infinite-numberof- agents case, we show that if the initial opinions follow a stable distribution, then the consensus also follows a stable distribution. In addition, we derive a closed-form expression of the probability density function of the consensus when the initial opinions follow a Gaussian distribution, a Cauchy distribution, or a L´evy distribution.

scholarly journals Colored HOMFLY-PT for hybrid weaving knot $$ {\hat{\mathrm{W}}}_3 $$(m, n)

2021 ◽  
Vol 2021 (6) ◽  
Author(s):  
Vivek Kumar Singh ◽  
Rama Mishra ◽  
P. Ramadevi
Keyword(s):  
Closed Form ◽  
Topological Strings ◽  
Chebyshev Polynomials ◽  
Fibonacci Numbers ◽  
Infinite Family ◽  
Closed Form Expression ◽  
Two Dimensional ◽  
Laurent Polynomials ◽  
Form Expression ◽  
Torus Knots

Abstract Weaving knots W(p, n) of type (p, n) denote an infinite family of hyperbolic knots which have not been addressed by the knot theorists as yet. Unlike the well known (p, n) torus knots, we do not have a closed-form expression for HOMFLY-PT and the colored HOMFLY-PT for W(p, n). In this paper, we confine to a hybrid generalization of W(3, n) which we denote as $$ {\hat{W}}_3 $$ W ̂ 3 (m, n) and obtain closed form expression for HOMFLY-PT using the Reshitikhin and Turaev method involving $$ \mathrm{\mathcal{R}} $$ ℛ -matrices. Further, we also compute [r]-colored HOMFLY-PT for W(3, n). Surprisingly, we observe that trace of the product of two dimensional $$ \hat{\mathrm{\mathcal{R}}} $$ ℛ ̂ -matrices can be written in terms of infinite family of Laurent polynomials $$ {\mathcal{V}}_{n,t}\left[q\right] $$ V n , t q whose absolute coefficients has interesting relation to the Fibonacci numbers $$ {\mathrm{\mathcal{F}}}_n $$ ℱ n . We also computed reformulated invariants and the BPS integers in the context of topological strings. From our analysis, we propose that certain refined BPS integers for weaving knot W(3, n) can be explicitly derived from the coefficients of Chebyshev polynomials of first kind.

Simple Closed-Form Expression to Estimate the Ring Ratio of 16-APSK to Compensate for Distortion in Nonlinear Systems

2017 ◽  
Author(s):  
M.J. Cañavate-Sánchez ◽  
A. Segneri ◽  
S. Godi ◽  
A. Georgiadis ◽  
S. Kosmopoulos ◽  
...  
Keyword(s):  
Nonlinear Systems ◽  
Closed Form ◽  
Closed Form Expression ◽  
Form Expression ◽  
Ring Ratio

Approximate closed-form expression for the ergodic capacity of polarisation-diversity MIMO systems

2004 ◽  
Vol 40 (19) ◽  
pp. 1192 ◽  
Author(s):  
J. Pérez ◽  
J. Ibáñez ◽  
L. Vielva ◽  
I. Santamaría
Keyword(s):  
Closed Form ◽  
Mimo Systems ◽  
Ergodic Capacity ◽  
Closed Form Expression ◽  
Form Expression
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