3.34. Hypergeometric Functions¶

The series:

$\sum_{k=0}^\infty t_k$

with $t_0=1$ is geometric if the ratio of two consecutive terms $t_{k+1}/t_k$ is a constant (with respect to $k$ ):

${t_{k+1} \over t_k} = x$

then we get:

$\sum_{k=0}^\infty t_k = \sum_{k=0}^\infty x^k$

It is hypergeometric if the ratio $t_{k+1}/t_k$ is a rational function (with respect to $k$ ):

${t_{k+1} \over t_k} = {P(k)\over Q(k)}$

where $P(k)$ and $Q(k)$ are polynomials in $k$ , which we can completely factor into the form

(3.34.1)¶ ${t_{k+1} \over t_k} = {P(k)\over Q(k)} = {(k+a_1)(k+a_2)\cdots(k+a_p)\over (k+b_1)(k+b_2)\cdots(k+b_q)(k+1)} x$

where $x$ is a constant and the $(k+1)$ factor is just a convention (if the polynomial $Q(k)$ does not contain the factor $(k+1)$ we can just add it to both numerator and denominator and absorb the “1” into $a_p$ ). The hypergeometric series is then given by:

${}_p F_q(a_1, a_2, \dots, a_p; b_1, b_2, \dots, b_q; x) = \sum_{k=0}^\infty {(a_1)_k (a_2)_k \cdots (a_p)_k \over (b_1)_k (b_2)_k \cdots (b_q)_k} {x^k\over k!}$

where

$(a)_k = {\Gamma(a+k)\over\Gamma(a)} = \begin{cases} a(a+1)(a+2)\cdots(a+k-1), & \mbox{if $k\ge 1$;} \\ 1, & \mbox{if $k=0$}\\ \end{cases}$

is the rising factorial function (also called the Pochhammer symbol).

To write a function as a hypergeometric series, we simply expand it in series and then write the ratio $t_{k+1}/t_k$ in the form (3.34.1) and immediately identify the proper ${}_p F_q$ function. If the ratio cannot be put into the form (3.34.1) then the function is not hypergeometric.

3.34.1. Convergence Conditions¶

If any $a_i=0, -1, -2, \dots$ , then the series is a polynomial of degree $-a_i$ .

If any $b_i=0, -1, -2, \dots$ then the denominators eventually become 0 (unless the series is terminated as a polynomial before that, due to the previous point) and the series is undefined.

Except the previous two cases, the radius of convergence $R$ of the hypergeometric series is:

$R = \begin{cases} \infty & \mbox{if $p \le q$;} \\ 1 & \mbox{if $p = q+1$;} \\ 0 & \mbox{if $p > q+1$.} \\ \end{cases}$

3.34.2. Elementary and Special Functions¶

The hypergeometric functions for low $p$ and $q$ have special names:

${}_0F_1$	confluent hypergeometric limit function
${}_1F_1$	Kummer’s confluent hypergeometric function of the first kind
${}_2F_1$	Gauss’ hypergeometric function

Most common functions can be expressed using ${}_p F_q$ as follows:

The Series 0F0¶

Elementary functions:

$e^{x} = \sum_{k=0}^\infty {x^k\over k!} = {}_0 F_0(x)$

The Series 1F0¶

Elementary functions:

${1\over 1-x} = \sum_{k=0}^\infty x^k = {}_1 F_0(1; x) {1\over (1-x)^a} = \sum_{k=0}^\infty {(a+k-1)!\over (a-1)! k!} x^k = {}_1 F_0(a; x) x^a = {}_1 F_0(-a; 1-x) \sqrt x = {}_1 F_0(-\half; 1-x)$

The Series 0F1¶

Elementary functions:

$\sin z = z \ {}_0F_1({\textstyle{3\over2}}; -{z^2\over 4}) \cos z = {}_0F_1(\half; -{z^2\over 4}) \sinh z = z \ {}_0F_1({\textstyle{3\over2}}; {z^2\over 4}) \cosh z = {}_0F_1(\half; {z^2\over 4})$

Bessel function:

$J_\alpha(x) = \sum_{k=0}^\infty {(-1)^k \left(x\over 2\right)^{2k+\alpha} \over k! (k+\alpha)!} = {\left(x\over2\right)^\alpha \over \Gamma(\alpha+1)} \ {}_0F_1\left(\alpha+1; -{x^2\over 4}\right)$

Spherical Bessel function of the first kind:

$j_\nu(x) = \sqrt{\pi\over 2x} J_{\nu+\half}(x) = {\sqrt\pi\left(x\over2\right)^\nu \over 2\Gamma(\nu+{3\over2})} \ {}_0F_1\left(\nu+{3\over2}; -{x^2\over 4}\right)$

Modified Bessel functions:

$I_\nu(z) = i^{-\nu} J_\nu(iz) = \sum_{k=0}^\infty {\left(x\over 2\right)^{2k+\nu} \over k! (k+\nu)!} = {1\over \Gamma(\nu+1)} \left(z\over 2\right)^\nu {}_0F_1\left(\nu+1; {z^2\over 4}\right) K_\nu(z) = {\Gamma(\nu)\over 2} \left(2\over z\right)^\nu {}_0F_1\left(1-\nu; {z^2\over 4}\right) + {\Gamma(-\nu)\over 2} \left(z\over 2\right)^\nu {}_0F_1\left(\nu+1; {z^2\over 4}\right)$

The Series 1F1¶

Elementary functions:

$z^a e^z = {}_1F_1(a; a-\half; -2z)$

Lower incomplete gamma function:

$\gamma(z, x) = x^z \Gamma(z) e^{-x} \sum_{k=0}^\infty {x^k\over \Gamma(z+k+1)} = x^z z^{-1} e^{-x}\ {}_1F_1(1; z+1; x) = x^z z^{-1}\ {}_1F_1(z; z+1; -x)$

Error function:

$\mbox{erf}(x) = {1\over\sqrt\pi}\gamma(\half, x^2) = {2x\over\sqrt\pi}\ {}_1F_1(\half; {\textstyle{3\over2}}, -x^2)$

Hermite polynomials:

$H_{2n}(x) = (-1)^n {(2n)!\over n!}\ {}_1F_1(-n;\half; x^2) H_{2n+1}(x) = (-1)^n {(2n+1)!\over n!}2x \ {}_1F_1(-n;{\textstyle{3\over2}}; x^2)$

Laguerre polynomials:

(3.34.2.1)¶ $L_n^\alpha(x) = \binom{n+\alpha}{n}\ {}_1F_1(-n;\alpha+1;x)$

Solution $P_{nl}(r)=r R_{nl}(r)$ of the radial Schrödinger equation in the Coulomb potential $V(r) = -{Z/r}$ (we use (3.34.2.1) in the second equation below):

$P_{nl}(r) = N_{nl} \left(2Zr\over n\right)^{l+1} e^{-{Zr\over n}} \ {}_1F_1\left(-n+l+1; 2l+2; {2Zr\over n}\right) = = N_{nl} \left(2Zr\over n\right)^{l+1} e^{-{Zr\over n}} \ L_{n-l-1}^{2l+1}\left({2Zr\over n}\right) {(2l+1)!(n-l-1)!\over (n+l)!} = = {1\over n} \sqrt{Z (n-l-1)! \over (n+l)!} \left(2Zr\over n\right)^{l+1} e^{-{Zr\over n}} \ L_{n-l-1}^{2l+1}\left({2Zr\over n}\right) N_{nl} = {1\over n(2l+1)!} \sqrt{Z(n+l)!\over (n-l-1)!}$

The Series 2F1¶

Elementary functions:

$\log(1+z) = z\ {}_2F_1(1, 1; 2; -z) \log(z) = (z-1)\ {}_2F_1(1, 1; 2; 1-z) \arcsin z = z\ {}_2F_1(\half, \half; {\textstyle{3\over2}}; z^2) \arccos z = {\pi\over2}-z\ {}_2F_1(\half, \half; {\textstyle{3\over2}}; z^2) \arctan z = z\ {}_2F_1(1, \half; {\textstyle{3\over2}}; -z^2)$

Legendre polynomials (and associated Legendre polynomials):

$P_n(z) = {}_2F_1\left(-n, n+1; 1; {1-z\over 2}\right) P_n^\mu(z) = {1\over\Gamma(1-\mu)} \left(1+z\over1-z\right)^{\mu\over2} {}_2F_1\left(-n, n+1; 1-\mu; {1-z\over 2}\right)$

Chebyshev polynomials:

$T_n(z) = {}_2F_1\left(-n, n;\half; {1-z\over 2}\right) U_n(z) = (n+1)\ {}_2F_1\left(-n, n+2;{\textstyle{3\over2}}; {1-z\over 2}\right)$

Gegenbauer polynomials:

$C_n^\alpha(z) = {(2\alpha)_n \over n!} \ {}_2F_1\left(-n, 2\alpha + n;\alpha+\half; {1-z\over 2}\right)$

Jacobi polynomials:

$P_n^{(\alpha, \beta)}(z) = {(\alpha+1)_n \over n!} \ {}_2F_1\left(-n, 1+\alpha+\beta+n;\alpha+1; {1-z\over 2}\right)$

Complete elliptic integrals:

$K(k) = {\pi\over 2}\ {}_2F_1( \half, \half; 1; k^2) E(k) = {\pi\over 2}\ {}_2F_1(-\half, \half; 1; k^2)$

The Series 3F2¶

Elementary functions:

$\tan(z) = {8z\over \pi^2-4z^2} \ {}_3F_2(1, \half-{z\over\pi}, \half + {z\over\pi}; {\textstyle{3\over2}}-{z\over\pi}, {\textstyle{3\over2}} + {z\over\pi}; 1)$

Dilogarithm:

$\mbox{Li}_2(z) = z\ {}_3F_2(1, 1, 1; 2, 2; z)$

Digamma:

$\psi(z) = (z-1)\ {}_3F_2(1, 1, 2-z; 2, 2; 1) -\gamma$

The Wigner 3j symbol:

$\begin{pmatrix} j_1 & j_2 & j_3 \\ m_1 & m_2 & m_3 \end{pmatrix} = (-1)^{-j_1 + j_2 + m_3} \delta_{-m_3, m_1+m_2} {1\over(-j_2+j_3+m_1)! (-j_1+j_3-m_2)!} {\sqrt{(j_1-j_2+j_3)! (-j_1+j_2+j_3)! (j_1+m_1)! (j_2-m_2)! (j_3+m_3)!(j_3-m_3)!}\over \sqrt{(j_1+j_2-j_3)!(j_1+j_2+j_3+1)!(j_1-m_1)!(j_2+m_2)!}} {}_3F_2(-j_1-j_2+j_3, m_1-j_1, -j_2-m_2; -j_1+j_3-m_2+1, -j_2+j_3+m_1+1; 1)$

The Series pFq¶

Polylogarithm:

$\mbox{Li}_s(z) = z\ {}_{s+1}F_s(1, 1, \dots, 1; 2, \dots, 2; z)$

Fermi-Dirac integral:

$I_\nu(x) = \int_0^\infty {t^\nu\over 1 + e^{t-x}} \d t = -\Gamma(\nu+1) \mbox{Li}_{\nu+1}(-e^x)$

3.34.3. Example I¶

By writing out the series expansion for the $t_{k+1}/t_k$ ratio we can prove that:

$p\ {}_1F_1(a; b; x) + q\ {}_1F_1(a+1; b; x) = (p+q)\ {}_2F_2\left(a, a\left({p\over q}+1\right)+1; b, a\left({p\over q}+1\right); x \right)$

The left hand side is equal to:

$p\ {}_1F_1(a; b; x) + q\ {}_1F_1(a+1; b; x) = \sum_{k=0}^\infty {p (a)_k + q(a+1)_k \over (b)_k k!} x^k$

We simplify the $t_k$ term:

$t_k = {p (a)_k + q(a+1)_k \over (b)_k k!} x^k = {(a)_k \left(p+q+{qk\over a}\right) \over (b)_k k!} x^k$

We calculate the ratio $t_{k+1}/t_k$ as well as $t_0$ to get the normalization:

$t_0 = p + q {t_{k+1}\over t_k} = {(k+a)\left(p+q+{q(k+1)\over a}\right) \over (k+b)(k+1) \left(p+q+{qk\over a}\right)} x = {(k+a)\left(k + a\left({p\over q}+1\right)+1\right) \over (k+b)\left(k + a\left({p\over q}+1\right)\right)(k+1)} x$

From which we read the arguments of the hypergeometric function ${}_2F_2$ on the right hand side and we need to multiply it by the normalization factor $t_0 = p+q$ .