Transformation formulas for the generalized hypergeometric function with integral parameter differences

Miller, Allen R.; Paris, R. B.

doi:10.1216/rmj-2013-43-1-291

Cited by 45 publications

(73 citation statements)

References 14 publications

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“…The differences between the

T - 1

upper and lower parameters of the hypergeometric function are the non‐negative integers

n_{1}, \dots, n_{T - 1}

and the results of Karlsson () and Miller and Paris (, ) relating to such functions can, therefore, be invoked. Specifically, the hypergeometric function embedded in

L_{1} false(λ, p false)

can be expressed in closed form as the product of the exponential of its argument

λ {false(1 - p false)}^{T}

and a polynomial of degree

m = \sum_{t = 1}^{T - 1} n_{t}

in that argument.…”

Section: Resultsmentioning

confidence: 99%

“…Specifically, the hypergeometric function embedded in

L_{1} false(λ, p false)

can be expressed in closed form as the product of the exponential of its argument

λ {false(1 - p false)}^{T}

and a polynomial of degree

m = \sum_{t = 1}^{T - 1} n_{t}

in that argument. The results based on the work of, Miller and Paris, ; Miller and Paris, are more algebraically tractable than those of Karlsson () and are used here. Thus, following Miller and Paris (, Theorem 2),

_{T - 1} F_{T - 1} ((|, \begin{matrix} center (n_{T} + 1) \\ center (n_{T} - n_{t} + 1) \end{matrix}) λ {(1 - p)}^{T}) = e^{λ {false(1 - p false)}^{T}} \sum_{k = 0}^{m} \frac{A_{k}}{A_{0}} {false[λ {false(1 - p false)}^{T} false]}^{k},

where, for

S false(j, k false)

a Stirling number of the second kind and

s_{m - j}

the coefficient of

x^{j}

in the expansion of

{false(n_{T} - n_{1} + 1 + x false)}_{n_{1}}, \dots {false(n_{T} - n_{T - 1} + 1 + x false)}_{n_{T - 1}}

\begin{matrix} right A_{k} & center = & left \sum_{j = k}^{m} S (j, k \end{matrix}

…”

Section: Resultsmentioning

confidence: 99%

“…Consider the N ‐mixture model for repeated count data with the unknown population size at a given site taken from a negative binomial distribution with mean

μ > 0

and shape or dispersion parameter

r > 0

(Royle, ). The likelihood can be derived either from first principles, using results from Miller and Paris (), or, following Dennis et al (), by taking the parameter

λ

in the likelihood associated with the Poisson mixing distribution to be gamma distributed and integrating appropriately. Full details of both approaches are given in Web Appendix B.…”

Section: Negative Binomialmentioning

confidence: 99%

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Maximum Likelihood Estimation for N-Mixture Models

Haines

2016

Biometrics

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The focus of this article is on the nature of the likelihood associated with N-mixture models for repeated count data. It is shown that the infinite sum embedded in the likelihood associated with the Poisson mixing distribution can be expressed in terms of a hypergeometric function and, thence, in closed form. The resultant expression for the likelihood can be readily computed to a high degree of accuracy and is algebraically tractable. Specifically, the likelihood equations can be simplified to some advantage, the concentrated likelihood in the probability of detection formulated and problematic cases identified. The results are illustrated by means of a simulation study and a real world example. The study is extended to N-mixture models with a negative binomial mixing distribution and results similar to those for the Poisson case obtained. N-mixture models with mixing distributions which accommodate excess zeros and, separately, with a beta-binomial distribution rather than a binomial used to model the intra-site counts are also investigated. However the results for these settings, while computationally attractive, do not provide insight into the nature of the maximum likelihood estimates.

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“…The differences between the

T - 1

upper and lower parameters of the hypergeometric function are the non‐negative integers

n_{1}, \dots, n_{T - 1}

and the results of Karlsson () and Miller and Paris (, ) relating to such functions can, therefore, be invoked. Specifically, the hypergeometric function embedded in

L_{1} false(λ, p false)

can be expressed in closed form as the product of the exponential of its argument

λ {false(1 - p false)}^{T}

and a polynomial of degree

m = \sum_{t = 1}^{T - 1} n_{t}

in that argument.…”

Section: Resultsmentioning

confidence: 99%

“…Specifically, the hypergeometric function embedded in

L_{1} false(λ, p false)

can be expressed in closed form as the product of the exponential of its argument

λ {false(1 - p false)}^{T}

and a polynomial of degree

m = \sum_{t = 1}^{T - 1} n_{t}

_{T - 1} F_{T - 1} ((|, \begin{matrix} center (n_{T} + 1) \\ center (n_{T} - n_{t} + 1) \end{matrix}) λ {(1 - p)}^{T}) = e^{λ {false(1 - p false)}^{T}} \sum_{k = 0}^{m} \frac{A_{k}}{A_{0}} {false[λ {false(1 - p false)}^{T} false]}^{k},

where, for

S false(j, k false)

a Stirling number of the second kind and

s_{m - j}

the coefficient of

x^{j}

in the expansion of

{false(n_{T} - n_{1} + 1 + x false)}_{n_{1}}, \dots {false(n_{T} - n_{T - 1} + 1 + x false)}_{n_{T - 1}}

\begin{matrix} right A_{k} & center = & left \sum_{j = k}^{m} S (j, k \end{matrix}

…”

Section: Resultsmentioning

confidence: 99%

“…Consider the N ‐mixture model for repeated count data with the unknown population size at a given site taken from a negative binomial distribution with mean

μ > 0

and shape or dispersion parameter

r > 0

(Royle, ). The likelihood can be derived either from first principles, using results from Miller and Paris (), or, following Dennis et al (), by taking the parameter

λ

in the likelihood associated with the Poisson mixing distribution to be gamma distributed and integrating appropriately. Full details of both approaches are given in Web Appendix B.…”

Section: Negative Binomialmentioning

confidence: 99%

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Maximum Likelihood Estimation for N-Mixture Models

Haines

2016

Biometrics

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“…Our version of the second Miller-Paris transformation (4) (see [20,Theorem 4],[14, Theorem 1]) is the following theorem.…”

Section: Corollarymentioning

confidence: 99%

“…Unlike Minton-Karlsson formulas dealing the generalized hypergeometric functions evaluated at 1 these transformations are certain identities for these functions evaluated at an arbitrary value of the argument. They were developed in a series of papers published over last 15 years, the most general form was presented in a seminal paper [20] by Miller and Paris. For a vector of positive integers m = (m 1 , .…”

Section: Introductionmentioning

confidence: 99%

Alternative Approach to Miller-Paris Transformations and Their Extensions

Karp

Prilepkina

2020

Trends in Mathematics

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Miller-Paris transformations are extensions of Euler's transformations for the Gauss hypergeometric functions to generalized hypergeometric functions of higher-order having integral parameter differences (IPD). In our recent work we computed the degenerate versions of these transformations corresponding to the case when one parameter difference is equal to a negative integer. The purpose of this paper is to present an independent new derivation of both the general and the degenerate forms of Miller-Paris transformations. In doing so we employ the generalized Stieltjes transform representation of the generalized hypergeometric functions and some partial fraction expansions. Our approach leads to different forms of the characteristic polynomials, one of them appears noticeably simpler than the original form due to Miller and Paris. We further present two extensions of the degenerate transformations to the generalized hypergeometric functions with additional free parameters and additional parameters with negative integral differences.

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An alternative proof of the extended Saalschütz summation theorem for the _{r + 3}F_{r + 2}(1) series with applications

Kim

Rathie

Paris

2015

Math Methods in App Sciences

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A simple proof is given of a new summation formula recently added in the literature for a terminating rC3 F rC2 .1/ hypergeometric series for the case when r pairs of numeratorial and denominatorial parameters differ by positive integers. This formula represents an extension of the well-known Saalschütz summation formula for a 3 F 2 .1/ series. Two applications of this extended summation formula are discussed. The first application extends two identities given by Ramanujan and the second, which also employs a similar extension of the Vandermonde-Chu summation theorem for the 2 F 1 series, extends certain reduction formulas for the Kampé de Fériet function of two variables given by Exton and Cvijović & Miller.

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Transformation formulas for the generalized hypergeometric function with integral parameter differences

Cited by 45 publications

References 14 publications

Maximum Likelihood Estimation for N-Mixture Models

Maximum Likelihood Estimation for N-Mixture Models

Alternative Approach to Miller-Paris Transformations and Their Extensions

An alternative proof of the extended Saalschütz summation theorem for the _{r + 3}F_{r + 2}(1) series with applications

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Transformation formulas for the generalized hypergeometric function with integral parameter differences

Cited by 45 publications

References 14 publications

Maximum Likelihood Estimation for N-Mixture Models

Maximum Likelihood Estimation for N-Mixture Models

Alternative Approach to Miller-Paris Transformations and Their Extensions

An alternative proof of the extended Saalschütz summation theorem for the r + 3Fr + 2(1) series with applications

Contact Info

Product

Resources

About

An alternative proof of the extended Saalschütz summation theorem for the _{r + 3}F_{r + 2}(1) series with applications