mathematics, a confluent hypergeometric function is a solution of a confluent hypergeometric equation, which is a degenerate form of a hypergeometric differential...

(Gaussian) hypergeometric function and the confluent hypergeometric function as special cases, which in turn have many particular special functions as special...

ordinary hypergeometric function 2F1(a,b;c;z) is a special function represented by the hypergeometric series, that includes many other special functions as...

Mittag-Leffler function, and can also be expressed as a confluent hypergeometric function (Kummer's function): erf ⁡ ( x ) = 2 x π M ( 1 2 , 3 2 , − x 2 ) . {\displaystyle...

of its kind: the generalized hypergeometric function and the MacRobert E-function had the same aim, but Meijer's G-function was able to include those as...

Coulomb potential and can be written in terms of confluent hypergeometric functions or Whittaker functions of imaginary argument. The Coulomb wave equation...

where 1F1 is the confluent hypergeometric function and J1 is the Bessel function of the first kind. Likewise the moment generating function can be calculated...

Confluent hypergeometric functions of the first kind. The conventional Hermite polynomials may also be expressed in terms of confluent hypergeometric...

mathematics, a Whittaker function is a special solution of Whittaker's equation, a modified form of the confluent hypergeometric equation introduced by...

; z ) {\displaystyle \;_{1}F_{1}(a;b;z)=M(a;b;z)} is the confluent hypergeometric function. Other pairs of independent solutions may be formed from linear...

{z^{s+k}}{s+k}}={\frac {z^{s}}{s}}M(s,s+1,-z),} where M is Kummer's confluent hypergeometric function. When the real part of z is positive, γ ( s , z ) = s − 1...

a=0.} Another connexion with the confluent hypergeometric functions is that E1 is an exponential times the function U(1,1,z): E 1 ( z ) = e − z U ( 1...

In probability theory and statistics, the moment-generating function of a real-valued random variable is an alternative specification of its probability...

mathematics, there are several functions known as Kummer's function. One is known as the confluent hypergeometric function of Kummer. Another one, defined...

characteristic function of the beta distribution to a Bessel function, since in the special case α + β = 2α the confluent hypergeometric function (of the first...

{i^{l}}{(m+nl+1)}}{\frac {x^{m+nl+1}}{l!}}} is a confluent hypergeometric function and also an incomplete gamma function ∫ x m e i x n d x = x m + 1 m + 1 1 F 1...

real-valued, Γ(x) is the gamma function and 1F1(a, b; x) is a confluent hypergeometric function. Some subfamilies of hypergeometric-Gaussian (HyGG) modes can...

stationary one-dimensional Schrödinger equation in terms of the confluent hypergeometric functions. The potential is given as V = V 0 1 + W ( e − x σ ) . {\displaystyle...

{1}{(1-t)^{\alpha +1}}}e^{-tx/(1-t)}.} Laguerre functions are defined by confluent hypergeometric functions and Kummer's transformation as L n ( α ) ( x...

which generalize Kummer's confluent hypergeometric function 1F1 of one variable and the confluent hypergeometric limit function 0F1 of one variable in a...

Confluence Project, a web-based volunteer project Confluent hypergeometric function, a mathematical function Confluent, a data streaming software company Convergence...

b ; z ) {\displaystyle M(a,b,z)=_{1}F_{1}(a;b;z)} is the confluent hypergeometric function of the first kind. When k is even, the raw moments become...

In mathematics, the Bateman function (or k-function) is a special case of the confluent hypergeometric function studied by Harry Bateman(1931). Bateman...

function Riesz function Hypergeometric functions: Versatile family of power series. Confluent hypergeometric function Associated Legendre functions Meijer G-function...

the plain and absolute moments can be expressed in terms of confluent hypergeometric functions 1 F 1 {\textstyle {}_{1}F_{1}} and U . {\textstyle U.} E ⁡...

the 24 symmetries of the hypergeometric differential equations obtained by Kummer. The symmetries fixing the local Heun function form a group of order 24...

, z ) {\displaystyle M(a,b,z)} is Kummer's confluent hypergeometric function. The characteristic function is given by: φ ( t ; k ) = M ( k 2 , 1 2 , −...

Horn function classification scheme. The total 34 Horn functions can be further categorised into 14 complete hypergeometric functions and 20 confluent hypergeometric...

-{\frac {d_{2}}{d_{1}}}\imath s\right)} where U(a, b, z) is the confluent hypergeometric function of the second kind. In instances where the F-distribution...

here by Cunningham (1908). It can be defined in terms of the confluent hypergeometric function U, by ω m , n ( x ) = e − x + π i ( m / 2 − n ) Γ ( 1 + n...