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

(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 (1) and immediately identify the proper {}_p F_q function. If the ratio cannot be put into the form (1) then the function is not hypergeometric.

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}

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:

(2)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 (2) 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)

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.

Example II

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

e^{-x}\ {}_1F_1(1; 2; 2x)
    = {}_0F_1\left({\textstyle{3\over 2}}; {x^2\over 4}\right)

We can also use the substitution z={x^2\over 4}:

e^{-2\sqrt z}\ {}_1F_1(1; 2; 4\sqrt z)
    = {}_0F_1\left({\textstyle{3\over 2}}; z\right)

Which is a special case of

{}_0F_1\left(a; z\right)
    = e^{-2\sqrt z}\ {}_1F_1(a-\half; 2a-1; 4\sqrt z)

for a={3\over 2}.

Example III

One way to express \sinh(z) is:

\sinh z = z e^{-z}\ {}_1F_1(1; 2; 2z)

using the previous example, this is equal to:

\sinh z
    = z e^{-z}\ {}_1F_1(1; 2; 2z)
    = z\ {}_0F_1\left({\textstyle{3\over 2}}; {z^2\over 4}\right)

So the lowest hypergeometric function that can express \sinh(z) is {}_0F_1.