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DLMF: §13.12 Products ‣ Kummer Functions ‣ Chapter 13 Confluent Hypergeometric Functions

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13 Confluent Hypergeometric Functions Kummer Functions 13.11 Series 13.13 Addition and Multiplication Theorems

§13.12 Products

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Keywords:: Kummer functions, products
Notes:: (13.12.1) follows from the fact that its left-hand side is bounded at infinity: use (13.2.39)–(13.2.41), and (13.7.3).
Permalink:: http://dlmf.nist.gov/13.12
See also:: Annotations for Ch.13

13.12.1		$M\left(a,b,z\right)M\left(-a,-b,-z\right)+\frac{a(a-b)z^{2}}{b^{2}(1-b^{2})}M% \left(1+a,2+b,z\right)M\left(1-a,2-b,-z\right)=1.$
ⓘ Symbols: $M\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : $={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ Kummer confluent hypergeometric function and $z$ : complex variable Referenced by: §13.12, §13.25 Permalink: http://dlmf.nist.gov/13.12.E1 Encodings: TeX, pMML, png See also: Annotations for §13.12 and Ch.13

For generalizations of this quadratic relation see Majima et al. (2000).

For integral representations, integrals, and series containing products of $M\left(a,b,z\right)$ and $U\left(a,b,z\right)$ see Erdélyi et al. (1953a, §6.15.3).

13.11 Series 13.13 Addition and Multiplication Theorems

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