Many of the special functions in mathematics are special cases of the
confluent hypergeometric function or the
hypergeometric function; see the corresponding articles for examples.
The series 0F0 As noted earlier, {}_0F_0(;;z) = e^z. The differential equation for this function is \frac{d}{dz}w = w, which has solutions w = ke^z where
k is a constant.
The series 0F1 The functions of the form {}_0F_1(;a;z) are called
confluent hypergeometric limit functions and are closely related to
Bessel functions. The relationship is: :J_\alpha(x)=\frac{(\tfrac{x}{2})^\alpha}{\Gamma(\alpha+1)} {}_0F_1\left (;\alpha+1; -\tfrac{1}{4}x^2 \right ). :I_\alpha(x)=\frac{(\tfrac{x}{2})^\alpha}{\Gamma(\alpha+1)} {}_0F_1\left (;\alpha+1; \tfrac{1}{4}x^2 \right ). The differential equation for this function is :w = \left (z\frac{d}{dz}+a \right )\frac{dw}{dz} or :z\frac{d^2w}{dz^2}+a\frac{dw}{dz}-w = 0. When
a is not a positive integer, the substitution :w = z^{1-a}u, gives a linearly independent solution :z^{1-a}\;{}_0F_1(;2-a;z), so the general solution is :k\;{}_0F_1(;a;z)+l z^{1-a}\;{}_0F_1(;2-a;z) where
k,
l are constants. (If
a is a positive integer, the independent solution is given by the appropriate Bessel function of the second kind.) A special case is: :{}_0F_1\left(;\frac{1}{2};-\frac{z^2}{4}\right)=\cos z
The series 1F0 An important case is: :{}_1F_0(a;;z) = (1-z)^{-a}. The differential equation for this function is :\frac{d}{dz}w =\left (z\frac{d}{dz}+a \right )w, or :(1-z)\frac{dw}{dz} = aw, which has solutions :w=k(1-z)^{-a} where
k is a constant. :{}_1F_0(1;;z) = \sum_{n \geqslant 0} z^n = (1-z)^{-1} is the
geometric series with ratio
z and coefficient 1. :z ~ {}_1F_0(2;;z) = \sum_{n \geqslant 0} n z^n = z (1-z)^{-2} is also useful.
The series 1F1 The functions of the form {}_1F_1(a;b;z) are called
confluent hypergeometric functions of the first kind, also written M(a;b;z). The incomplete gamma function \gamma(a,z) is a special case. The differential equation for this function is :\left (z\frac{d}{dz}+a \right )w = \left (z\frac{d}{dz}+b \right )\frac{dw}{dz} or :z\frac{d^2w}{dz^2}+(b-z)\frac{dw}{dz}-aw = 0. When
b is not a positive integer, the substitution :w = z^{1-b}u, gives a linearly independent solution :z^{1-b}\;{}_1F_1(1+a-b;2-b;z), so the general solution is :k\;{}_1F_1(a;b;z)+l z^{1-b}\;{}_1F_1(1+a-b;2-b;z) where
k,
l are constants. When a is a non-positive integer, −
n, {}_1F_1(-n;b;z) is a polynomial. Up to constant factors, these are the
Laguerre polynomials. This implies
Hermite polynomials can be expressed in terms of 1
F1 as well.
The series 1F2 Relations to other functions are known for certain parameter combinations only. The function x\; {}_1F_2\left(\frac{1}{2};\frac{3}{2},\frac{3}{2};-\frac{x^2}{4}\right) is the
antiderivative of the
cardinal sine. With modified values of a_1 and b_1, one obtains the antiderivative of \sin(x^\beta)/x^\alpha. The
Lommel function is s_{\mu, \nu} (z) = \frac{z^{\mu + 1}}{(\mu - \nu + 1)(\mu + \nu + 1)} {}_1F_2\left(1; \frac{\mu}{2} - \frac{\nu}{2} + \frac{3}{2} , \frac{\mu}{2} + \frac{\nu}{2} + \frac{3}{2} ;-\frac{z^2}{4}\right) .
The series 2F0 The confluent hypergeometric function of the second kind can be written as: :U(a,b,z) = z^{-a} \; {}_2 F_0 \left( a, a-b+1; ; -\frac{1}{z}\right).
The series 2F1 Historically, the most important are the functions of the form {}_2F_1(a,b;c;z). These are sometimes called '''Gauss's hypergeometric functions
, classical standard hypergeometric or often simply hypergeometric functions. The term Generalized hypergeometric function'
is used for the functions p'
F'q'' if there is risk of confusion. This function was first studied in detail by
Carl Friedrich Gauss, who explored the conditions for its convergence. The differential equation for this function is : \left (z\frac{d}{dz}+a \right ) \left (z\frac{d}{dz}+b \right )w =\left (z\frac{d}{dz}+c \right )\frac{dw}{dz} or :z(1-z)\frac {d^2w}{dz^2} + \left[c-(a+b+1)z \right] \frac {dw}{dz} - ab\,w = 0. It is known as the
hypergeometric differential equation. When
c is not a positive integer, the substitution :w = z^{1-c}u gives a linearly independent solution : z^{1-c}\; {}_2F_1(1+a-c,1+b-c;2-c;z), so the general solution for |
z| k\; {}_2F_1(a,b;c;z)+l z^{1-c}\; {}_2F_1(1+a-c,1+b-c;2-c;z) where
k,
l are constants. Different solutions can be derived for other values of
z. In fact there are 24 solutions, known as the
Kummer solutions, derivable using various identities, valid in different regions of the complex plane. When
a is a non-positive integer, −
n, :{}_2F_1(-n,b;c;z) is a polynomial. Up to constant factors and scaling, these are the
Jacobi polynomials. Several other classes of orthogonal polynomials, up to constant factors, are special cases of Jacobi polynomials, so these can be expressed using 2
F1 as well. This includes
Legendre polynomials and
Chebyshev polynomials. A wide range of integrals of elementary functions can be expressed using the hypergeometric function, e.g.: :\int_0^x\sqrt{1+y^\alpha}\,\mathrm{d}y=\frac{x}{2+\alpha}\left \{\alpha\;{}_2F_1\left(\tfrac{1}{\alpha},\tfrac{1}{2};1+\tfrac{1}{\alpha};-x^\alpha \right) +2\sqrt{x^\alpha+1} \right \},\qquad \alpha\neq0.
The series 2F2 The hypergeometric series {}_2F_2 is generally associated with integrals of products of power functions and the exponential function. As such, the
exponential integral can be written as: :\operatorname{Ei}(x)=x{}_2F_2(1,1;2,2;x)+\ln x+\gamma.
The series 3F0 The
Mott polynomials can be written as: :s_n(x)=(-x/2)^n{}_3F_0(-n,\frac{1-n}{2},1-\frac{n}{2};;-\frac{4}{x^2}).
The series 3F2 The function ::\operatorname{Li}_2(x) = \sum_{n>0}\,{x^n}{n^{-2}} = x \; {}_3F_2(1,1,1;2,2;x) is the
dilogarithm The function ::Q_n(x;a,b,N)= {}_3F_2(-n,-x,n+a+b+1;a+1,-N+1;1) is a
Hahn polynomial.
The series 4F3 The function ::p_n(t^2)=(a+b)_n(a+c)_n(a+d)_n \; {}_4F_3\left( -n, a+b+c+d+n-1, a-t, a+t ; a+b, a+c, a+d ;1\right) is a
Wilson polynomial. All roots of a
quintic equation can be expressed in terms of radicals and the
Bring radical, which is the real solution to x^5 + x + a = 0. The Bring radical can be written as: ::\operatorname{BR}(a) = -a \; {}_4F_3\left( \frac{1}{5}, \frac{2}{5}, \frac{3}{5}, \frac{4}{5} ; \frac{1}{2}, \frac{3}{4}, \frac{5}{4} ; -\frac{3125a^4}{256} \right). The partition function Z(K) of the 2D isotropic
Ising model with no external magnetic field was found by
Onsager in the 1940s and can be expressed as ::\ln Z(K) = \ln(2 \cosh 2K) -k^2 {}_4F_3\left( 1, 1, \frac{3}{2}, \frac{3}{2} ; 2, 2, 2; 16k^2 \right), with K=\frac{J}{k_\mathrm{B}T} and k=\frac{1}{2}\tanh 2K\,\operatorname{sech} 2K.
The series q+1Fq The functions ::\operatorname{Li}_q(z)=z \; {}_{q+1}F_q\left(1,1,\ldots,1;2,2,\ldots,2;z\right) ::\operatorname{Li}_{-p}(z)=z \; {}_pF_{p-1}\left(2,2,\ldots,2;1,1,\ldots,1;z\right) for q\in\mathbb{N}_0 and p\in\mathbb{N} are the
Polylogarithm. For each integer
n≥2, the roots of the polynomial
xn−
x+t can be expressed as a sum of at most
N−1 hypergeometric functions of type
n+1F
n, which can always be reduced by eliminating at least one pair of
a and
b parameters. ==Generalizations==