hypergeom

Hypergeometric function

Description

example

hypergeom(a,b,z) represents the generalized hypergeometric function.

Examples

Hypergeometric Function for Numeric and Symbolic Arguments

Depending on whether the input is floating point or symbolic, hypergeom returns floating point or symbolic results.

Compute the hypergeometric function for these numbers. Because these numbers are floating point, hypergeom returns floating-point results.

A = [hypergeom([1 2], 2.5, 2),...
hypergeom(1/3, [2 3], pi),...
hypergeom([1 1/2], 1/3, 3*i)]
A =
-1.2174 - 0.8330i   1.2091 + 0.0000i  -0.2028 + 0.2405i

Return exact symbolic results by converting at least one of the inputs to symbolic form by using sym. For most symbolic (exact) inputs, hypergeom returns unresolved symbolic calls.

symA = [hypergeom([1 2], 2.5, sym(2)),...
hypergeom(1/3, [2 3], sym(pi)),...
hypergeom([1 1/2], sym(1/3), 3*i)]
symA =
[ hypergeom([1, 2], 5/2, 2), hypergeom(1/3, [2, 3], pi), hypergeom([1/2, 1], 1/3, 3i)]

Convert the symbolic result to high-precision floating-point by using vpa.

vpa(symA)
ans =
[ - 1.2174189301051728850455150601879 - 0.83304055090469367131547768563638i,...
1.2090631887094273193917339575087,...
- 0.20275169745081962937527290365593 + 0.24050134226872040357481317881983i]

Special Values of Hypergeometric Function

Show that hypergeom returns special values for certain input values.

syms a b c d x
hypergeom([], [], x)
ans =
exp(x)
hypergeom([a b c d], [a b c d], x)
ans =
exp(x)
hypergeom(a, [], x)
ans =
1/(1 - x)^a

Show that the hypergeometric function is always 1 at 0.

syms a b c d
hypergeom([a b], [c d], 0)
ans =
1

If, after cancelling identical parameters in the first two arguments, the list of upper parameters contains 0, the resulting hypergeometric function is constant with the value 1. For details, see Algorithms.

hypergeom([0 0 2 3], [a 0 4], x)
ans =
1

If, after canceling identical parameters in the first two arguments, the upper parameters contain a negative integer larger than the largest negative integer in the lower parameters, the hypergeometric function is a polynomial.

hypergeom([-4 -2 3], [-3 1 4], x)
ans =
(3*x^2)/5 - 2*x + 1

Hypergeometric functions reduce to other special functions for certain input values.

hypergeom([1], [a], x)
hypergeom([a], [a, b], x)
ans =
(exp(x/2)*whittakerM(1 - a/2, a/2 - 1/2, -x))/(-x)^(a/2)

ans =
x^(1/2 - b/2)*gamma(b)*besseli(b - 1, 2*x^(1/2))

Handling Expressions That Contain Hypergeometric Functions

Many symbolic functions, such as diff and taylor, handle expressions containing hypergeom.

Differentiate this expression containing the hypergeometric function.

syms a b c d x
diff(1/x*hypergeom([a b],[c d],x), x)
ans =
(a*b*hypergeom([a + 1, b + 1], [c + 1, d + 1], x))/(c*d*x)...
- hypergeom([a, b], [c, d], x)/x^2

Compute the Taylor series of this hypergeometric function.

taylor(hypergeom([1 2],3,x), x)
ans =
(2*x^5)/7 + x^4/3 + (2*x^3)/5 + x^2/2 + (2*x)/3 + 1

Input Arguments

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Upper parameters of hypergeometric function, specified as a number, variable, symbolic expression, symbolic function, or vector.

Lower parameters of hypergeometric function, specified as a number, variable, symbolic expression, symbolic function, or vector.

Input, specified as a number, vector, matrix, or array, or a symbolic number, variable, array, function, or expression.

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Generalized Hypergeometric Function

The generalized hypergeometric function of order p, q is defined as follows.

${}_{p}F{}_{q}\left(a;b;z\right)={}_{p}F{}_{q}\left({a}_{1},\dots ,{a}_{j},\dots ,{a}_{p};{b}_{1},\dots ,{b}_{k},\dots ,{b}_{q};z\right)=\sum _{n=0}^{\infty }\left(\frac{{\left({a}_{1}\right)}_{n}\dots {\left({a}_{j}\right)}_{n}\dots {\left({a}_{p}\right)}_{n}}{{\left({b}_{1}\right)}_{n}\dots {\left({b}_{k}\right)}_{n}\dots {\left({b}_{q}\right)}_{n}}\right)\left(\frac{{z}^{n}}{n!}\right).$

Here a = [a1,a2,...,ap] and b = [b1,b2,...,bq] are vectors of lengths p and q, respectively.

(a)k and (b)k are Pochhammer symbols.

For empty vectors a and b, hypergeom is defined as follows.

$\begin{array}{l}{}_{0}F{}_{q}\left(;b;z\right)=\sum _{k=0}^{\infty }\frac{1}{{\left({b}_{1}\right)}_{k}{\left({b}_{2}\right)}_{k}\dots {\left({b}_{q}\right)}_{k}}\left(\frac{{z}^{k}}{k!}\right)\\ {}_{p}F{}_{0}\left(a;;z\right)=\sum _{k=0}^{\infty }{\left({a}_{1}\right)}_{k}{\left({a}_{2}\right)}_{k}\dots {\left({a}_{p}\right)}_{k}\left(\frac{{z}^{k}}{k!}\right)\\ {}_{0}F{}_{0}\left(;;z\right)=\sum _{k=0}^{\infty }\left(\frac{{z}^{k}}{k!}\right)={e}^{z}.\end{array}$

Pochhammer Symbol

${\left(x\right)}_{n}=\frac{\Gamma \left(x+n\right)}{\Gamma \left(x\right)}.$

If n is a positive integer, then (x)n = x(x + 1)...(x + n - 1).

Algorithms

The hypergeometric function is

${}_{p}F{}_{q}\left(a;b;z\right)={}_{p}F{}_{q}\left({a}_{1},\dots ,{a}_{j},\dots ,{a}_{p};{b}_{1},\dots ,{b}_{k},\dots ,{b}_{q};z\right)=\sum _{n=0}^{\infty }\left(\frac{{\left({a}_{1}\right)}_{n}\dots {\left({a}_{j}\right)}_{n}\dots {\left({a}_{p}\right)}_{n}}{{\left({b}_{1}\right)}_{n}\dots {\left({b}_{k}\right)}_{n}\dots {\left({b}_{q}\right)}_{n}}\right)\left(\frac{{z}^{n}}{n!}\right).$

• The hypergeometric function has convergence criteria:

• Converges if p ≤ q and |z| < ∞.

• Converges if p = q + 1 and |z| < 1. For |z| >= 1, the series diverges, and is defined by analytic continuation.

• Diverges if p > q + 1 and z ≠ 0. Here, the series is defined by an asymptotic expansion of pFq(a;b;z) around z = 0. The branch cut is the positive real axis.

• The function is a polynomial, called the hypergeometric polynomial, if any aj is a nonpositive integer.

• The function is undefined:

• If any bk is a nonpositive integer such that bk > aj where aj is also a nonpositive integer, because division by 0 occurs

• If any bk is a nonpositive integer and no aj is a nonpositive integer

• The function has reduced order when upper and lower parameter values are equal and cancel. If r values of the upper and lower parameters are equal (that is, a = [a1,…,ap - r, c1,…,cr], b = [b1,…,bq - r, c1,…,cr]), then the order (p, q) of pFq(a;b;z) is reduced to (p - r, q - r):

$\begin{array}{l}{}_{p}F{}_{q}\left({a}_{1},\dots ,{a}_{p-r},{c}_{1},\dots ,{c}_{r};{b}_{1},\dots ,{b}_{q-r},{c}_{1},\dots ,{c}_{r};z\right)=\\ {}_{p-r}F{}_{q-r}\left({a}_{1},\dots ,{a}_{p-r};{b}_{1},\dots ,{b}_{q-r};z\right)\end{array}$

This rule applies even if any ci is zero or a negative integer [2].

• pFq(a;b;z) is symmetric. That is, it does not depend on the order a1, a2, … in a or b1, b2, … in b.

• $U\left(z\right)={}_{p}F{}_{q}\left(a;b;z\right)$ satisfies the differential equation in z

Here, (δ + a) represents

$\prod _{i=1}^{p}\left(\delta +{a}_{i}\right).$

And (δ + b) represents

$\prod _{j=1}^{q}\left(\delta +{b}_{j}\right).$

Thus, the order of this differential equation is max(p, q + 1), and the hypergeometric function is only one of its solutions. If p < q + 1, this differential equation has a regular singularity at z = 0 and an irregular singularity at z = ∞. If p = q + 1, the points z = 0, z = 1, and z = ∞ are regular singularities, which explains the convergence properties of the hypergeometric series.

• The hypergeometric function has these special values:

• pFp(a;a;z) = 0F0(;;z) = ez.

• pFq(a;b;z) = 1 if the list of upper parameters a contains more 0s than the list of lower parameters b.

• pFq(a;b;0) = 1.

References

[1] Oberhettinger, F. “Hypergeometric Functions.” Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. (M. Abramowitz and I. A. Stegun, eds.). New York: Dover, 1972.

[2] Luke, Y.L. "The Special Functions and Their Approximations", Vol. 1, Academic Press, New York, 1969.

[3] Prudnikov, A.P., Yu.A. Brychkov, and O.I. Marichev, "Integrals and Series", Vol. 3: More Special Functions, Gordon and Breach, 1990.

Version History

Introduced before R2006a