Multi-Center Integrals from the General Expansion Formula Developed Previously for Slater Orbitals

1982 ◽  
pp. 35-51
Author(s):  
R. R. Sharma
Keyword(s):  
Expansion Formula ◽  
Slater Orbitals ◽  
General Expansion

A General Expansion Formula

1918 ◽  
Vol 51 (1) ◽  
pp. 38-44
Author(s):  
G. J. L.
Keyword(s):  
Expansion Formula ◽  
General Expansion

scholarly journals Note on the Heaviside Expansion Formula

1931 ◽  
Vol 17 (12) ◽  
pp. 678-684 ◽  
Author(s):  
J. M. D. Valle
Keyword(s):  
Expansion Formula

Slater orbitals for the atoms of transition elements

1983 ◽  
Vol 23 (5) ◽  
pp. 795-796
Author(s):  
I. A. Morev ◽  
E. P. Smirnov
Keyword(s):  
Transition Elements ◽  
Slater Orbitals

scholarly journals Befriending Askey–Wilson polynomials

2014 ◽  
Vol 17 (03) ◽  
pp. 1450015 ◽  
Author(s):  
Paweł J. Szabłowski
Keyword(s):  
Characteristic Function ◽  
Hermite Polynomials ◽  
Probabilistic Interpretation ◽  
Conjugate Pair ◽  
Linear Combinations ◽  
Expansion Formula ◽  
Wilson Polynomials ◽  
The Way

We recall five families of polynomials constituting a part of the so-called Askey–Wilson scheme. We do this to expose properties of the Askey–Wilson (AW) polynomials that constitute the last, most complicated element of this scheme. In doing so we express AW density as a product of the density that makes q-Hermite polynomials orthogonal times a product of four characteristic function of q-Hermite polynomials (2.9) just pawing the way to a generalization of AW integral. Our main results concentrate mostly on the complex parameters case forming conjugate pairs. We present new fascinating symmetries between the variables and some newly defined (by the appropriate conjugate pair) parameters. In particular in (3.12) we generalize substantially famous Poisson–Mehler expansion formula (3.16) in which q-Hermite polynomials are replaced by Al-Salam–Chihara polynomials. Further we express Askey–Wilson polynomials as linear combinations of Al-Salam–Chihara (ASC) polynomials. As a by-product we get useful identities involving ASC polynomials. Finally by certain re-scaling of variables and parameters we reach AW polynomials and AW densities that have clear probabilistic interpretation.

On a General Expansion in the Theory of the Tides

1929 ◽  
Vol s2-29 (1) ◽  
pp. 527-536 ◽  
Author(s):  
J. Proudman
Keyword(s):  
General Expansion

Correction of the Kneser-Sommerfeld Expansion Formula

1981 ◽  
Vol 50 (4) ◽  
pp. 1391-1391 ◽  
Author(s):  
Hiroshi Hayashi
Keyword(s):  
Expansion Formula

scholarly journals Comultiplication Rules for the Double Schur Functions and Cauchy Identities

10.37236/102 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
A. I. Molev
Keyword(s):  
Symmetric Functions ◽  
Schur Functions ◽  
Primitive Elements ◽  
Expansion Formula ◽  
New Family ◽  
Dual Object ◽  
Multiplication Rule ◽  
Distinguished Basis ◽  
Natural Way

The double Schur functions form a distinguished basis of the ring $\Lambda(x\!\parallel\!a)$ which is a multiparameter generalization of the ring of symmetric functions $\Lambda(x)$. The canonical comultiplication on $\Lambda(x)$ is extended to $\Lambda(x\!\parallel\!a)$ in a natural way so that the double power sums symmetric functions are primitive elements. We calculate the dual Littlewood–Richardson coefficients in two different ways thus providing comultiplication rules for the double Schur functions. We also prove multiparameter analogues of the Cauchy identity. A new family of Schur type functions plays the role of a dual object in the identities. We describe some properties of these dual Schur functions including a combinatorial presentation and an expansion formula in terms of the ordinary Schur functions. The dual Littlewood–Richardson coefficients provide a multiplication rule for the dual Schur functions.

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