Depending on its arguments, hypergeom can return floating-point or exact symbolic results.
Compute the hypergeometric function for these numbers. Because these numbers are not symbolic objects, you get 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
Compute the hypergeometric function for the numbers converted to symbolic objects. For most symbolic (exact) numbers, 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], 3*i)
Use vpa to approximate symbolic results with the required number of digits:
ans = - 1.21741893 - 0.8330405509*i 1.209063189 - 0.2027516975 + 0.2405013423*i
The hypergeometric function has special values for some parameters:
syms a b c d x hypergeom(, , x) hypergeom([a, b, c, d], [a, b, c, d], x) hypergeom(a, , x)
ans = exp(x) ans = exp(x) ans = 1/(1 - x)^a
Any hypergeometric function, evaluated at 0, has the value 1:
syms a b c d hypergeom([a, b], [c, d], 0)
ans = 1
If, after canceling identical parameters, the list of upper parameters contains 0, the resulting hypergeometric function is constant with the value 1:
hypergeom([0, 0, 2, 3], [a, 0, 4], x)
ans = 1
If, after canceling identical parameters, the upper parameters contain a negative integer larger than the largest negative integer in the lower parameters, the hypergeometric function is a polynomial. If all parameters as well as the argument x are numeric, a corresponding explicit value is returned:
hypergeom([(-4), -2 , 3], [-3, 1, 4], x*pi*sqrt(2))
ans = (6*pi^2*x^2)/5 - 2*2^(1/2)*pi*x + 1
Hypergeometric functions also reduce to other special functions for some parameters:
hypergeom(, [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))
Many functions, such as diff and taylor, can handle expressions containing hypergeom.
Differentiate this expression containing 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], , x), x)
ans = (2*x^5)/7 + x^4/3 + (2*x^3)/5 + x^2/2 + (2*x)/3 + 1
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.
Argument of hypergeometric function, specified as a number, variable, symbolic expression, symbolic function, or vector. If z is a vector, hypergeom(a,b,z) is evaluated element-wise.
The generalized hypergeometric function of order p, q is defined as follows:
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:
The Pochhammer symbol is defined as follows:
If n is a positive integer, then (x)n = x(x + 1)...(x + n - 1).
For most exact arguments, the hypergeometric function returns a symbolic function call. If an upper parameter coincides with a lower parameter, these values cancel and are removed from the parameter lists.
If, after cancellation of identical parameters, the upper parameters contain a negative integer larger than the largest negative integer in the lower parameters, then pFq(a;b;z) is a polynomial in z.
The following special values are implemented:
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.
Converges for any |z| < ∞ if p ≤ q.
Converges for |z| < 1 if p = q + 1. For |z| >= 1, the series diverges, and pFq is defined by analytic continuation.
Diverges for any z <> 0 if p > q + 1. The series defines an asymptotic expansion of pFq(a;b;z) around z = 0. The positive real axis is the branch cut.
If one of the parameters in a is equal to 0 or a negative integer, then the series terminates, turning into what is called a hypergeometric polynomial.
pFq(a;b;z) is symmetric with respect to the parameters, that is, it does not depend on the order of the arrangement a1, a2, … in a or b1, b2, … in b.
If at least one upper parameter equals n = 0,-1,-2,…, the function turns into a polynomial of degree n. If the previous condition for the lower parameters b is relaxed, and there is some lower parameter equal to m = 0,-1,-2,…, then the function pFq(a;b;z) also reduces to a polynomial in z provided n > m. It is undefined if m > n or if no upper parameter is a nonpositive integer (resulting in division by 0 in one of the series coefficients). The case m = n is handled by the following rule.
. If for r values of the upper parameters, there are r values of the lower parameters equal to them (that is, a = [a1,…,ap - r, c1,…,cr], b = [b1,…,bq - r, c1,…,cr]), then the order (p, q) of the function pFq(a;b;z) is reduced to (p - r, q - r):
This rule applies even if any of the ci is zero or a negative integer. For details, see Luke, Y.L. "The Special Functions and Their Approximations", vol. 1, p. 42.
satisfies a differential equation in z:
where (δ + a) and (δ + b) stand for
respectively. 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, thus explaining the convergence properties of the hypergeometric series.
 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.
 Luke, Y.L. "The Special Functions and Their Approximations", Vol. 1, Academic Press, New York, 1969.
 Prudnikov, A.P., Yu.A. Brychkov, and O.I. Marichev, "Integrals and Series", Vol. 3: More Special Functions, Gordon and Breach, 1990.