On a summation formula for the Appell function F2

1967 ◽  
Vol 63 (4) ◽  
pp. 1087-1089 ◽  
Author(s):  
H. M. Srivastava
Keyword(s):  
Quantum Mechanics ◽  
Exact Solutions ◽  
Double Series ◽  
Summation Formula ◽  
Systematic Analysis ◽  
Image Position ◽  
Appell Function

1. In the course of a systematic analysis of certain problems in quantum mechanics it has been observed that their exact solutions can be expressed in terms of the Appell function F2 defined by means of (see e.g. (8), p. 211)where, as usual,and for convergence of the double series,

On Appell's function F2

1969 ◽  
Vol 65 (3) ◽  
pp. 687-689
Author(s):  
H. L. Manocha
Keyword(s):  
Quantum Mechanics ◽  
Exact Solutions ◽  
Summation Formula ◽  
Image Position ◽  
The Right

1. It has been observed that certain problems in quantum mechanics have their exact solutions which can be expressed in terms of Appell's function F2 as defined by (e.g. (7), p. 211)Having regard to this fact, Srivastava (5) proved a summation formulawhere R(λ) > 1, R(α) > − 1 and x⇌y indicates the presence on the right of a second term that originates from the first by interchanging x and y.

An infinite summation formula associated with Appell's function F2

1969 ◽  
Vol 65 (3) ◽  
pp. 679-682 ◽  
Author(s):  
H. M. Srivastava
Keyword(s):  
Double Series ◽  
Summation Formula ◽  
Image Position ◽  
Appell Function

1. Recently in these proceedings we proved that, if ℜ(α) > 1 and ℜ(α) > − 1, then ((10), p. 1088)where F2 denotes the Appell function (see, e.g., (8), p. 211),with, as usual,and for covergence of the double series,and x⇌y indicates the presence of a second term that originates from the first by interchanging x and y.

Addendum to ‘A hypergeometric transformation associated with the Appell function F4’

1969 ◽  
Vol 66 (3) ◽  
pp. 569-570
Author(s):  
H. M. Srivastava
Keyword(s):  
Hypergeometric Function ◽  
Double Series ◽  
The One ◽  
Image Position ◽  
Appell Function

The hypergeometric transformation formulae (6) and the one that follows on p. 766 in my paper (1) should be read asandrespectively, where 2F1[a,b; c; z] denotes Gauss's hypergeometric function, and ((2), p. 211)with, as usual,the conditions of convergence for the double series being

On the Properties of an Entire Function of Two Complex Variables

1968 ◽  
Vol 20 ◽  
pp. 51-57
Author(s):  
Arun Kumar Agarwal
Keyword(s):  
Entire Function ◽  
Double Series ◽  
Complex Variables ◽  
Maximum Term ◽  
Image Position

1. Letbe an entire function of two complex variables z1 and z2, holomorphic in the closed polydisk . LetFollowing S. K. Bose (1, pp. 214-215), μ(r1, r2; ƒ ) denotes the maximum term in the double series (1.1) for given values of r1 and r2 and v1{m2; r1, r2) or v1(r1, r2), r2 fixed, v2(m1, r1, r2) or v2(r1, r2), r1 fixed and v(r1r2) denote the ranks of the maximum term of the double series (1.1).

scholarly journals Some Finite Analogues of the Poisson Summation Formula

1961 ◽  
Vol 12 (3) ◽  
pp. 133-138 ◽  
Author(s):  
L. Carlitz
Keyword(s):  
Summation Formula ◽  
Poisson Summation Formula ◽  
Euler’S Constant ◽  
Image Position ◽  
Positive Integers ◽  
Euler's Constant ◽  
Poisson Summation

1. Guinand (2) has obtained finite identities of the typewhere m, n, N are positive integers and eitherorwhere γ is Euler's constant and the notation ∑′ indicates that when x is integral the term r = x is multiplied by ½. Clearly there is no loss of generality in taking N = 1 in (1.1).

Adiabatic second-order energy derivatives in quantum mechanics

1958 ◽  
Vol 54 (2) ◽  
pp. 251-257 ◽  
Author(s):  
W. Byers Brown ◽  
H. C. Longuet Higgins
Keyword(s):  
Quantum Mechanics ◽  
General Equation ◽  
Canonical Ensemble ◽  
Second Order ◽  
Order Derivative ◽  
Interacting Particles ◽  
Energy Derivatives ◽  
Order Energy ◽  
Classical Analogue ◽  
Image Position

ABSTRACTThe general equation for the adiabatic second-order derivative of the energy En of an eigenstate with respect to parameters λ and λ′ occurring in the Hamiltonian ℋ isThe applications of this equation to molecules (λ, λ′ = nuclear position coordinates) and to enclosed assemblies of interacting particles (λ = λ′ = volume) are discussed, and the classical analogue of the equation for a micro-canonical ensemble is derived.

scholarly journals A Novel Exact Solution of the 2+1-Dimensional Radial Dirac Equation for the Generalized Dirac Oscillator with the Inverse Potentials

2019 ◽  
Vol 2019 ◽  
pp. 1-11 ◽  
Author(s):  
ZiLong Zhao ◽  
ZhengWen Long ◽  
MengYao Zhang
Keyword(s):  
Exact Solution ◽  
Quantum Mechanics ◽  
Dirac Equation ◽  
Exact Solutions ◽  
Bethe Ansatz ◽  
Dirac Oscillator ◽  
Solvable Models ◽  
Ansatz Method ◽  
Radial Dirac Equation

The generalized Dirac oscillator as one of the exact solvable models in quantum mechanics was introduced in 2+1-dimensional world in this paper. What is more, the general expressions of the exact solutions for these models with the inverse cubic, quartic, quintic, and sixth power potentials in radial Dirac equation were further given by means of the Bethe ansatz method. And finally, the corresponding exact solutions in this paper were further discussed.

scholarly journals Certain Expansions in the Algebra of Quantum Mechanics

1932 ◽  
Vol 3 (2) ◽  
pp. 118-127 ◽  
Author(s):  
Neal H. McCoy
Keyword(s):  
Quantum Mechanics ◽  
Complex Scalar ◽  
Scalar Constant ◽  
Image Position

§ 1. Introduction. The algebra of quantum mechanics is characterized by the fact that the variables p, q obey all the laws of ordinary algebra except that multiplication is non-commutative and instead there exists a relation of the formwhere c is a real or complex scalar constant and is thus commutative with both p and q.

scholarly journals Ramanujan's remarkable summation formula and an interesting convolution identity

1993 ◽  
Vol 47 (1) ◽  
pp. 155-162 ◽  
Author(s):  
S. Bhargava Chandrashekar Adiga ◽  
D.D. Somashekara
Keyword(s):  
Summation Formula ◽  
Special Cases ◽  
Image Position

In this note we obtain a convolution identity for the coefficients Bn(α, θ, q) defined byusing Ramanujan's 1Ψ1 summation. The identity contains as special cases convolution identities of Kung-Wei Yang and a few more interesting analogue.

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