New generating functions for Jacobi and related polynomials

Srivástava, H. M.; Singhal, J. P.

doi:10.1016/0022-247x(73)90244-8

Cited by 37 publications

(11 citation statements)

References 1 publication

Supporting

Mentioning

Contrasting

Order By: Relevance

“…following the method originally developed in [32]. We omit the details, since we only wish to use the result.…”

Section: Proof Of the Construction For F M (Z)mentioning

confidence: 99%

“…In order to obtain the asymptotics, we will need to interpret equation (134) in terms of equation (133), again following Srivastava and Singhal [32]. In the case of ψ L , we should let (see [10])…”

Section: Proof Of the Construction For F M (Z)mentioning

confidence: 99%

“…For the rest of this section we will briefly discuss Lagrange inversion and show how to derive the integral representations that Carteret et al used for the ψ-function to derive uniformly convergent asymptotics. Chen and Ismail's work on the Jacobi polynomials [12] uses the generating function for J (γ+aj,β+bj) j (0) of Srivastava and Singhal [32] which in our notation becomes …”

Section: Outline Of the Proof For G M (Z)mentioning

confidence: 99%

See 2 more Smart Citations

Evanescence in coined quantum walks

Carteret¹,

Richmond²,

Temme³

2005

J. Phys. A: Math. Gen.

View full text Add to dashboard Cite

C e n t r u m v o o r W i s k u n d e e n I n f o r m a t i c a MAS Modelling, Analysis and Simulation Modelling, Analysis and SimulationEvanescence in coined quantum walks ABSTRACT In this paper we complete the analysis begun by two of the authors in a previous work on the discrete quantum walk on the line [J. Phys. A 36:8775-8795 (2003) quant-ph/0303105]. We obtain uniformly convergent asymptotics for the "exponential decay'' regions at the leading edges of the main peaks in the Schrödinger (or wave-mechanics) picture. This calculation required us to generalise the method of stationary phase and we describe this extension in some detail, including self-contained proofs of all the technical lemmas required. We also rigorously establish the exact Feynman equivalence between the path-integral and wavemechanics representations for this system using some techniques from the theory of special functions. Taken together with the previous work, we can now prove every theorem by both routes. AbstractIn this paper we complete the analysis begun by two of the authors in a previous work on the discrete quantum walk on the infinite line [J. Phys. A 36:8775-8795 (2003); quant-ph/0303105]. We obtain uniformly convergent asymptotics for the "exponential decay" regions at the leading edges of the main peaks in the Schrödinger (or wave-mechanics) picture. This calculation required us to generalise the method of stationary phase and we describe this extension in some detail, including self-contained proofs of all the technical lemmas required. We also rigorously establish the exact Feynman equivalence between the path-integral and wave-mechanics representations for this system using some techniques from the theory of special functions. Taken together with the previous work, we can now prove every theorem by both routes.

show abstract

“…following the method originally developed in [32]. We omit the details, since we only wish to use the result.…”

Section: Proof Of the Construction For F M (Z)mentioning

confidence: 99%

Section: Proof Of the Construction For F M (Z)mentioning

confidence: 99%

Section: Outline Of the Proof For G M (Z)mentioning

confidence: 99%

See 1 more Smart Citation

Evanescence in coined quantum walks

Carteret¹,

Richmond²,

Temme³

2005

J. Phys. A: Math. Gen.

View full text Add to dashboard Cite

show abstract

“…A subtle error was uncovered in the derivation of the asymptotics obtained by applying Darboux's method [7,9] to the Jacobi polynomial generating function of Srivastava and Singhal [8]. In Darboux's method, the asymptotic growth rate is determined by the location of the singularity nearest to the origin of the generating function.…”

Section: Introductionmentioning

confidence: 99%

Refined estimates on the growth rate of Jacobi polynomials

Izen

2007

Journal of Approximation Theory

View full text Add to dashboard Cite

The growth of the Jacobi polynomials P (k, ) n (1 − 2 2 ) is studied as n and k are simultaneously increased along lines in the kn plane with n integral. In particular, we apply a correction to a result of Chen and Ismail to show that for 0 < < 1, an P ( +an, ) n (1 − 2 2 ) decays exponentially as n → ∞ when a > 2 /(1 − ). For −1 < a < 2 /(1 − ), the decay is shown to be O(n −1/2 ).

show abstract

“…In 1977, motivated by the work of Srivastava and Singhal [27] on the Jacobi polynomials, Carlitz are general one-and two-parameter coefficients. Srivastava [23] proposed a generalization of Carlitz's theorem for the sequence of functions f …”

Section: Introductionmentioning

confidence: 99%