scholarly journals An electrostatic interpretation of the zeros of sieved ultraspherical polynomials

2020 ◽  
Vol 61 (5) ◽  
pp. 053501
Author(s):  
K. Castillo ◽  
M. N. de Jesus ◽  
J. Petronilho
Keyword(s):  
Ultraspherical Polynomials ◽  
Electrostatic Interpretation

scholarly journals New Specific and General Linearization Formulas of Some Classes of Jacobi Polynomials

Mathematics ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 74
Author(s):  
Waleed Mohamed Abd-Elhameed ◽  
Afnan Ali
Keyword(s):  
Chebyshev Polynomials ◽  
Hypergeometric Functions ◽  
Jacobi Polynomials ◽  
Algebraic Computation ◽  
Ultraspherical Polynomials ◽  
Current Article ◽  
Linearization Coefficients

The main purpose of the current article is to develop new specific and general linearization formulas of some classes of Jacobi polynomials. The basic idea behind the derivation of these formulas is based on reducing the linearization coefficients which are represented in terms of the Kampé de Fériet function for some particular choices of the involved parameters. In some cases, the required reduction is performed with the aid of some standard reduction formulas for certain hypergeometric functions of unit argument, while, in other cases, the reduction cannot be done via standard formulas, so we resort to certain symbolic algebraic computation, and specifically the algorithms of Zeilberger, Petkovsek, and van Hoeij. Some new linearization formulas of ultraspherical polynomials and third-and fourth-kinds Chebyshev polynomials are established.

Positivity of basic sums of ultraspherical polynomials

Analysis ◽  
1998 ◽  
Vol 18 (4) ◽  
pp. 313-332 ◽  
Author(s):  
Gavin Brown ◽  
Stamatis Koumandos ◽  
Kun-Yang Wang
Keyword(s):  
Ultraspherical Polynomials

The electrostatic calculation of molecular energies - IV. Optimum paired-electron orbitals and the electrostatic method

1956 ◽  
Vol 235 (1201) ◽  
pp. 224-234 ◽  
Keyword(s):  
Simple Procedure ◽  
Wave Functions ◽  
Optimum Form ◽  
Method Of Calculation ◽  
Hartree Fock ◽  
Electron Orbital ◽  
Electrostatic Method ◽  
Valence States ◽  
Paired Electron ◽  
Electrostatic Interpretation

Equations which determine the optimum form of paired-electron orbitals are derived. It is shown that for large nuclear separations these equations become the Hartree-Fock equa­tions for appropriate valence states of the separated atoms. An electrostatic interpretation of chemical bonding is developed using optimum paired-electron orbital functions. For these wave functions this simple procedure yields results identical with those obtained by the conventional method of calculation based on the Hamiltonian integral. Numerical computations by the electrostatic method are also discussed.

Some Characterizations of the Ultraspherical Polynomials

1968 ◽  
Vol 11 (3) ◽  
pp. 457-464 ◽  
Author(s):  
N.A. Al-Salam ◽  
W. A. Al-Salam
Keyword(s):  
Ultraspherical Polynomial ◽  
Ultraspherical Polynomials ◽  
Exact Degree ◽  
Image Position

Let be the nth ultraspherical polynomial. Also let . The following generating relation is well known (3, p.98).It can also be written as1.1This suggests the consideration of the class of polynomial sets {Qn(x), n = 0, 1, 2,…}, Qn(x) is of exact degree n and1.2

A Relation Between Ultraspherical and Jacobi Polynomial Sets

1953 ◽  
Vol 5 ◽  
pp. 301-305 ◽  
Author(s):  
Fred Brafman
Keyword(s):  
Jacobi Polynomial ◽  
Legendre Polynomials ◽  
Jacobi Polynomials ◽  
Ultraspherical Polynomials ◽  
Image Position ◽  
Special Case

The Jacobi polynomials may be defined bywhere (a)n = a (a + 1) … (a + n — 1). Putting β = α gives the ultraspherical polynomials which have as a special case the Legendre polynomials .

Convexity of the largest zero of the ultraspherical polynomials

1996 ◽  
Vol 4 (3) ◽  
pp. 275-278 ◽  
Author(s):  
C.G. Kokologiannaki ◽  
P.D. Siafarikas
Keyword(s):  
Ultraspherical Polynomials
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