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Dougall's Formula (other)
Dougall's formula may refer to one of two formulas for hypergeometric series, both named after John Dougall: *Dougall's formula for the sum of a 7''F''6 hypergeometric series *Dougall's formula for the sum of a bilateral hypergeometric series In mathematics, a bilateral hypergeometric series is a series Σ''a'n'' summed over ''all'' integers ''n'', and such that the ratio :''a'n''/''a'n''+1 of two terms is a rational function of ''n''. The definition of the generalized hyper ... {{mathdab Hypergeometric functions ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Dougall (mathematician)
Dr John Dougall FRSE (June 1867 – 24 February 1960) was "one of Scotland's leading mathematicians".. Two formulas are named Dougall's formula after him: one for the sum of a 7''F''6 hypergeometric series, and another for the sum of a bilateral hypergeometric series. Life Dougall was born in June 1867 in Kippen, a small village near Stirling, Scotland; his father, a watchmaker and postmaster, had nine children, among whom John was the eldest.. He was educated locally at Kippen School. He left school at age 13 to become a post office worker, but a year later entered Glasgow University, from which he earned an M.A. in 1886. (He was later given a doctorate by the same university.) After graduating, he taught mathematics at the Glasgow and West of Scotland Technical College before becoming an editor and translator of mathematical publications for Blackie and Son, a Glasgow publisher. He died on 24 February 1960 in Glasgow. Dougall became a member of the Edinburgh Mathematical So ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalized Hypergeometric Function
In mathematics, a generalized hypergeometric series is a power series in which the ratio of successive coefficients indexed by ''n'' is a rational function of ''n''. The series, if convergent, defines a generalized hypergeometric function, which may then be defined over a wider domain of the argument by analytic continuation. The generalized hypergeometric series is sometimes just called the hypergeometric series, though this term also sometimes just refers to the Gaussian hypergeometric series. Generalized hypergeometric functions include the (Gaussian) hypergeometric function and the confluent hypergeometric function as special cases, which in turn have many particular special functions as special cases, such as elementary functions, Bessel functions, and the classical orthogonal polynomials. Notation A hypergeometric series is formally defined as a power series :\beta_0 + \beta_1 z + \beta_2 z^2 + \dots = \sum_ \beta_n z^n in which the ratio of successive coefficients ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bilateral Hypergeometric Series
In mathematics, a bilateral hypergeometric series is a series Σ''a''''n'' summed over ''all'' integers ''n'', and such that the ratio :''a''''n''/''a''''n''+1 of two terms is a rational function of ''n''. The definition of the generalized hypergeometric series is similar, except that the terms with negative ''n'' must vanish; the bilateral series will in general have infinite numbers of non-zero terms for both positive and negative ''n''. The bilateral hypergeometric series fails to converge for most rational functions, though it can be analytically continued to a function defined for most rational functions. There are several summation formulas giving its values for special values where it does converge. Definition The bilateral hypergeometric series ''p''H''p'' is defined by :_pH_p(a_1,\ldots,a_p;b_1,\ldots,b_p;z)= _pH_p\left(\begina_1&\ldots&a_p\\b_1&\ldots&b_p\\ \end;z\right)= \sum_^\infty \fracz^n where :(a)_n=a(a+1)(a+2)\cdots(a+n-1)\, is the rising factorial or Pochha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |