Confluent hypergeometric function: Difference between revisions

Revision as of 15:36, 3 August 2024

Revision as of 19:59, 11 August 2024

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===Other equations===

Confluent Hypergeometric Functions can be used to solve the Extended Confluent Hypergeometric Equation whose general form is given as:

:<math>z\frac{d^2w}{dz^2} +(b-z)\frac{dw}{dz} -\left(\sum_{m=0}^M a_m z^m\right)w = 0</math> <ref>{{cite journal|last1=Campos|first1=L.M.B.C.|title=On Some Solutions of the Extended Confluent Hypergeometric Differential Equation|journal=Journal of Computational and Applied Mathematics|year=2001|volume=137|number=1|doi=10.1016/s0377-0427(00)00706-8|pages=177–200|bibcode=2001JCoAM.137..177C |mr=1865885}}</ref>

Note that for {{math|''M'' {{=}} 0}} or when the summation involves just one term, it reduces to the conventional Confluent Hypergeometric Equation.

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==Application to continued fractions==

By applying a limiting argument to [[Gauss's continued fraction]] it can be shown that<ref>{{cite journal|first1=Evelyn|last1=Frank | year=1956|title=A new class of continued fraction expansions for the ratios of hypergeometric functions| journal=Trans. Am. Math. Soc.|volume=81|number=2|pages=453-476453–476|mr= 0076937|jstor=1992927|doi=10.1090/S0002-9947-1956-0076937-0}}</ref>

:<math>\frac{M(a+1,b+1,z)}{M(a,b,z)} = \cfrac{1}{1 - \cfrac{{\displaystyle\frac{b-a}{b(b+1)}z}}

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* {{cite journal | last= Tricomi | first= Francesco G. | author-link= Francesco Giacomo Tricomi | title= Sulle funzioni ipergeometriche confluenti | language= it | journal= Annali di Matematica Pura ed Applicata |series=Series 4 | year= 1947 | volume= 26 | pages= 141–175 | issn= 0003-4622 | mr= 0029451 | doi=10.1007/bf02415375| s2cid= 119860549 | doi-access= free }}

* {{cite book | last= Tricomi | first= Francesco G. | title= Funzioni ipergeometriche confluenti | language= it | ___location= Rome | publisher= Edizioni cremonese | year= 1954 | series= Consiglio Nazionale Delle Ricerche Monografie Matematiche | volume= 1 | isbn= 978-88-7083-449-9 | mr= 0076936}}

* {{cite book | last1=Oldham | first1=K.B. | last2=Myland | first2=J. | last3=Spanier | first3=J. | title=An Atlas of Functions: with Equator, the Atlas Function Calculator | publisher=Springer New York | series=An Atlas of Functions | year=2010 | isbn=978-0-387-48807-3 | url=https://books.google.com/books?id=UrSnNeJW10YC&pg=PA75 | access-date=2017-08-23}}

==External links==