Generalized Hypergeometric Function - Convergence Conditions

Convergence Conditions

There are certain values of the a_j and b_k for which the numerator or the denominator of the coefficients is 0.

• If any a_j is a non-positive integer (0, −1, −2, etc.) then the series only has a finite number of terms and is, in fact, a polynomial of degree −a_j.
• If any b_k is a non-positive integer (excepting the previous case with −b_k < a_j) then the denominators become 0 and the series is undefined.

Excluding these cases, the ratio test can be applied to determine the radius of convergence.

• If p < q + 1 then the ratio of coefficients tends to zero. This implies that the series converges for any finite value of z. An example is the power series for the exponential function.
• If p = q + 1 then the ratio of coefficients tends to one. This implies that the series converges for |z| < 1 and diverges for |z| > 1. Whether it converges for |z| = 1 is more difficult to determine. Analytic continuation can be employed for larger values of z.
• If p > q + 1 then the ratio of coefficients grows without bound. This implies that, besides z = 0, the series diverges. This is then a divergent or asymptotic series, or it can be interpreted as a symbolic shorthand for a differential equation that the sum satisfies.

The question of convergence for p=q+1 when z is on the unit circle is more difficult. It can be shown that the series converges absolutely at z=1 if

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