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Continued Fraction Representations for Functions Related to the Gamma Function
| Content Provider | Scilit |
|---|---|
| Author | Lange, L. J. |
| Copyright Year | 2020 |
| Abstract | Our fundamental goal in this work is to present and justify a variety of representations for a multitude of complex functions related to the gamma function Γ(z). For each function considered we give an explicit representation of it as a Laplace transform of another function, as the limit of a series or product of rational functions, and (most importantly to us) as the limit of one or more continued fractions. A functional equation that it satisfies is also associated with each function studied. With regard to the validity of the representations presented, we have primarily concerned ourselves with studying the convergence behavior and justifying the validity of the continued fraction (c.f.) expansions. We feel that another important contribution we have made in the continued fraction category is that for at least one continued fraction associated with each included special function we give an explicit formula for the modifying sequence ${r_{n}$} used to determine a Bauer-Muir transform (see Theorem 3.4 for the definition) of the continued fraction. In light of the work of Jacobsen (1986) on the theory of general convergence, the sequences ${r_{n}$} (which lead to a sequence of modified approximants for the given c.f.) could prove valuable in future studies on the speed of convergence of c.f. expansions. We use a technique to establish convergence which we shall refer to as the Bauer-Muir transform method. Simply put, this method amounts to the following: We first establish that a continued fraction K associated with a given function F converges on some subset S of the real line. Next we compute a transform K ^ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072591/cdfacc39-b3df-40a4-a844-eb5c89fe0656/content/eq3320.tif"/> of K and use known results to establish that K and K ^ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072591/cdfacc39-b3df-40a4-a844-eb5c89fe0656/content/eq3321.tif"/> converge to the same value K(z) on S. But in our 234cases, this implies that K(z) satisfies a solvable functional relation. This leads to a series or product representation for K(z) and to the conclusion that K ( z ) = F ( z ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072591/cdfacc39-b3df-40a4-a844-eb5c89fe0656/content/eq3322.tif"/> on S. Finally, it is established that these c.f. and non c.f. representations have analytic extensions to a domain D in the complex plane containing S, so that K ( z ) = F ( z ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072591/cdfacc39-b3df-40a4-a844-eb5c89fe0656/content/eq3323.tif"/> on D by the Identity Theorem for analytic functions. |
| Related Links | https://content.taylorfrancis.com/books/download?dac=C2006-0-16181-4&isbn=9781003072591&format=googlePreviewPdf |
| Ending Page | 279 |
| Page Count | 47 |
| Starting Page | 233 |
| DOI | 10.1201/9781003072591-11 |
| Language | English |
| Publisher | Informa UK Limited |
| Publisher Date | 2020-12-17 |
| Access Restriction | Open |
| Subject Keyword | Book Name: Continued Fractions and Orthogonal Functions Series Or Product Continued Fraction Functions Related Theorem Satisfies Representations Behavior |
| Content Type | Text |
| Resource Type | Chapter |
