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Relevant bibliographies by topics / Hypergeometric Kummer's function

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Academic literature on the topic 'Hypergeometric Kummer's function'

Author: Grafiati

Published: 4 June 2025

Last updated: 23 June 2025

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Journal articles on the topic "Hypergeometric Kummer's function"

1

Qureshi, M. I., M. Sadiq Khan, M. A. Pathan, and N. U. Khan. "Some multivariable Gaussian hypergeometric extensions of the Preece theorem." ANZIAM Journal 48, no. 1 (2006): 143–50. http://dx.doi.org/10.1017/s1446181100003473.

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AbstractSome generalisations of the Preece theorem involving the product of two Kummer's functions 1F1 are obtained using Dixon's theorem and some well-known identities. Its special cases yield various new transformations and reduction formulae involving Pathan's quadruple hypergeometric function and Srivastava's quadruple hypergeometric function F(4) and triple hypergeometric function F(3). Some known results of Preece, Pathan and Bailey are also obtained as special cases.

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2

Milovanovic, Gradimir, Rakesh Parmar, and Arjun Rathie. "A study of generalized summation theorems for the series 2F1 with an applications to Laplace transforms of convolution type integrals involving Kummer's functions 1F1." Applicable Analysis and Discrete Mathematics 12, no. 1 (2018): 257–72. http://dx.doi.org/10.2298/aadm171017002m.

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Motivated by recent generalizations of classical theorems for the series 2F1 [Integral Transform. Spec. Funct. 229(11), (2011), 823-840] and interesting Laplace transforms of Kummer's confluent hypergeometric functions obtained by Kim et al. [Math. Comput. Modelling 55 (2012), 1068-1071], first we express generalized summations theorems in explicit forms and then by employing these, we derive various new and useful Laplace transforms of convolution type integrals by using product theorem of the Laplace transforms for a pair of Kummer's confluent hypergeometric function.

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3

Mathews, W. N., M. A. Esrick, Z. Y. Teoh, and J. K. Freericks. "A physicist's guide to the solution of Kummer's equation and confluent hypergeometric functions." Condensed Matter Physics 25, no. 3 (2022): 33203. http://dx.doi.org/10.5488/cmp.25.33203.

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The confluent hypergeometric equation, also known as Kummer's equation, is one of the most important differential equations in physics, chemistry, and engineering. Its two power series solutions are the Kummer function, M(a,b,z), often referred to as the confluent hypergeometric function of the first kind, and M ≡ z1-bM(1+a-b, 2-b,z), where a and b are parameters that appear in the differential equation. A third function, the Tricomi function, U(a,b,z), sometimes referred to as the confluent hypergeometric function of the second kind, is also a solution of the confluent hypergeometric equation that is routinely used. Contrary to common procedure, all three of these functions (and more) must be considered in a search for the two linearly independent solutions of the confluent hypergeometric equation. There are situations, when a, b, and a - b are integers, where one of these functions is not defined, or two of the functions are not linearly independent, or one of the linearly independent solutions of the differential equation is different from these three functions. Many of these special cases correspond precisely to cases needed to solve problems in physics. This leads to significant confusion about how to work with confluent hypergeometric equations, in spite of authoritative references such as the NIST Digital Library of Mathematical Functions. Here, we carefully describe all of the different cases one has to consider and what the explicit formulas are for the two linearly independent solutions of the confluent hypergeometric equation. The procedure to properly solve the confluent hypergeometric equation is summarized in a convenient table. As an example, we use these solutions to study the bound states of the hydrogenic atom, correcting the standard treatment in textbooks. We also briefly consider the cutoff Coulomb potential. We hope that this guide will aid physicists to properly solve problems that involve the confluent hypergeometric differential equation.

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4

Morita, Tohru. "Solution of Inhomogeneous Differential Equations with Polynomial Coefficients in Nonstandard Analysis, in Terms of the Green's Function." Journal of Advances in Mathematics and Computer Science 39, no. 1 (2024): 20–28. http://dx.doi.org/10.9734/jamcs/2024/v39i11859.

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Discussions are presented by Morita and Sato on the problem of obtaining the particular solution of an inhomogeneous differential equation with polynomial coefficients in terms of the Green's function. In a preceding paper, solution is given without using the Green's function, on the basis of nonstandard analysis, for a restricted class of inhomogeneous terms. In the present paper, the corresponding solutions are given in terms of the Green's function. It is applied to Kummer's and the hypergeometric differential equation.

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5

Mohammed, Asmaa Orabi, Medhat A. Rakha, Mohammed M. Awad, and Arjun K. Rathie. "On several new Laplace transforms of generalized hypergeometric functions 2F2(x)." Boletim da Sociedade Paranaense de Matemática 39, no. 4 (2021): 97–109. http://dx.doi.org/10.5269/bspm.42207.

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By employing generalizations of Gauss's second, Bailey's and Kummer's summation theorems obtained earlier by Rakha and Rathie, we aim to establish unknown Laplace transform of six rather general formulas of generalized hypergeometric function 2F2[a,b;c,d;x]. The results obtained in this paper are simple, interesting, easily established and may be useful in theoretical physics, engineering and mathematics. Results obtained earlier by Kim et al. and Choi and Rathie follow special cases of our main findings.

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6

Moothathu, T. S. K. "The Best Estimators of Quantiles and the Three Means of the Pareto Distribution." Calcutta Statistical Association Bulletin 35, no. 3-4 (1986): 111–22. http://dx.doi.org/10.1177/0008068319860301.

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In this paper we develop the uniformly minimum variance unbiased (best) estimators of the quantiles, mean, geometric mean and harmonic mean of the Pareto distribuion in the case when both the shape parameter a and the scale parameter k are unknown and in cases when one of them alone is unknown. The best estimates are in terms of the Bessel Function o F1 and Kummer's function 1 F1. The variance of the best estimator are found out, which are in terms of F2 , the Appell function of second kind and ψ2 , a confluent hypergeometric function of two variables. Further we prove that every best estimator of this paper is strongly consistent.

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7

Guillemin, Fabrice, and Didier Pinchon. "Continued Fraction Analysis of the Duration of an Excursion in an M/M/∞ System." Journal of Applied Probability 35, no. 1 (1998): 165–83. http://dx.doi.org/10.1239/jap/1032192560.

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We show in this paper how the Laplace transform θ* of the duration θ of an excursion by the occupation process {Λt} of an M/M/∞ system above a given threshold can be obtained by means of continued fraction analysis. The representation of θ* by a continued fraction is established and the [m−1/m] Padé approximants are computed by means of well known orthogonal polynomials, namely associated Charlier polynomials. It turns out that the continued fraction considered is an S fraction and as a consequence the Stieltjes transform of some spectral measure. Then, using classic asymptotic expansion properties of hypergeometric functions, the representation of the Laplace transform θ* by means of Kummer's function is obtained. This allows us to recover an earlier result obtained via complex analysis and the use of the strong Markov property satisfied by the occupation process {Λt}. The continued fraction representation enables us to further characterize the distribution of the random variable θ.

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8

Guillemin, Fabrice, and Didier Pinchon. "Continued Fraction Analysis of the Duration of an Excursion in an M/M/∞ System." Journal of Applied Probability 35, no. 01 (1998): 165–83. http://dx.doi.org/10.1017/s0021900200014765.

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We show in this paper how the Laplace transform θ* of the duration θ of an excursion by the occupation process {Λ t } of an M/M/∞ system above a given threshold can be obtained by means of continued fraction analysis. The representation of θ* by a continued fraction is established and the [m−1/m] Padé approximants are computed by means of well known orthogonal polynomials, namely associated Charlier polynomials. It turns out that the continued fraction considered is an S fraction and as a consequence the Stieltjes transform of some spectral measure. Then, using classic asymptotic expansion properties of hypergeometric functions, the representation of the Laplace transform θ* by means of Kummer's function is obtained. This allows us to recover an earlier result obtained via complex analysis and the use of the strong Markov property satisfied by the occupation process {Λ t }. The continued fraction representation enables us to further characterize the distribution of the random variable θ.

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9

Poudel, Madhav Prasad, Harsh Vardhan Harsh, Narayan Prasad Pahari, and Dinesh Panthi. "Kummer’s Theorems, Popular Solutions and Connecting Formulas on Hypergeometric Function." Journal of Nepal Mathematical Society 6, no. 1 (2023): 48–56. http://dx.doi.org/10.3126/jnms.v6i1.57413.

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The hypergeometric series is an extension of the geometric series. The confluent hypergeometric function is the solution of the hypergeometric differential equation [θ(θ +b−1)−z(θ +a)]w = 0. Kummer’s first formula and Kummer’s second formula are of significant importance in solving the hypergeometric differential equations. Kummer has developed six solutions for the differential equation and twenty connecting formulas during the period of 1865-1866. Each connecting formula consist of a solution expressed as the combination of two other solutions. Recently in 2021, these solutions were extensively used by Schweizer [13] in practical problems specially in Physics. Here we extend the connecting formulas obtained by Kummer to obtain the other six solutions w1(z), w2(z), w3(z), w4(z), w5(z) and w6(z) as the combination of three solutions.

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10

Yuriy, Khomyak, Naumenko Ievgeniia, Zheglova Victoriia, and Popov Vadim. "MINIMIZING THE MASS OF A FLAT BOTTOM OF CYLINDRICAL APPARATUS." Eastern-European Journal of Enterprise Technologies 2, no. 1 (92) (2018): 42–50. https://doi.org/10.15587/1729-4061.2018.126141.

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In the bodies of cylindrical apparatuses that operate under pressure, one of the weak elements is a flat bottom whose thickness is increased by 4…5 times in comparison with the wall thickness. This is due to the fact that the bottom is exposed to a more unfavorable bending deformation compared to the wall that «works» on stretching. In order to reduce specific metal consumption for the bottom, we propose the optimization of the shape of a radial cross-section by a rational redistribution of the material: to increase thickness of the bottom in the region of its contact with the wall and to significantly reduce it in the central zone. To describe a variable thickness of the bottom, we applied the Gauss equation with an arbitrary parameter that determines the intensity of change in the thickness in radial direction. We have obtained a general solution to the differential equation of the problem on bending a bottom at a given law of change in its thickness, which is represented using the hypergeometric Kummer’s functions. A technique for concretizing the resulting solution was proposed and implemented, based on the application of conditions of contact between a cylindrical shell and a bottom. The solution derived was used to minimize the mass of the bottom. We have designed a zone of transition from the bottom to the wall whose strength was verified by the method of finite elements under actual conditions

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Book chapters on the topic "Hypergeometric Kummer's function"

1

Duverney, Daniel. "Kummer Hypergeometric Function." In An Introduction to Hypergeometric Functions. Springer International Publishing, 2024. http://dx.doi.org/10.1007/978-3-031-65144-1_6.

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2

Takayama, Nobuki. "Generating Kummer Type Formulas for Hypergeometric Functions." In Algebra, Geometry and Software Systems. Springer Berlin Heidelberg, 2003. http://dx.doi.org/10.1007/978-3-662-05148-1_7.

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3

Andrews, Larry C. "The Confluent Hypergeometric Functions." In Special Functions Of Mathematics For Engineers. Oxford University PressOxford, 1998. http://dx.doi.org/10.1093/oso/9780198565581.003.0010.

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Abstract Whereas Gauss was largely responsible for the systematic study of the hypergeometric function, E. E. Kummer (1810-1893) is the person most associated with developing properties of the related confluent hypergeometric function. Kummer published his work on this function in 1836,* and since that time it has been commonly referred to as Kummer’s function. Like the hypergeometric function, the confluent hypergeometric function is related to a large number of other functions.

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4

Yang, Xiao-Jun. "Hypergeometric supertrigonometric and superhyperbolic functions via Kummer confluent hypergeometric series." In An Introduction to Hypergeometric, Supertrigonometric, and Superhyperbolic Functions. Elsevier, 2021. http://dx.doi.org/10.1016/b978-0-12-824154-7.00010-6.

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5

Srinivasa Rao, K. "Group theory of the Kummer solutions of the Gauss differential equation." In Generalized Hypergeometric Functions: Transformations and group theoretical aspects. IOP Publishing, 2018. http://dx.doi.org/10.1088/978-0-7503-1496-1ch3.

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