Consider the set of all generalized hypergeometric functions. I am trying to figure out which operations this set is closed under. For example, is the sum of two generalized hypergeometric functions equal to another generalized hypergeometric function? I am most interested in the "common" operations: addition, subtraction, multiplication, division, composition, differentiation, and antidifferentiation. I am sure that someone has written a paper on this topic, but there is so much information about the generalized hypergeometric function that it is hard to work through it all. I would appreciate any information or links on the subject. Thanks!
Asked
Active
Viewed 103 times
1
-
By "generalized hypergeometric function" do you mean a ${}_pF_q$ function or something else? There are many possible generalizations. – Robert Israel Jul 23 '15 at 18:30
-
Yes, that's the version I was thinking of. Thanks for your answer. – Joel Turnblade Jul 23 '15 at 19:19
1 Answers
4
If by "generalized hypergeometric function" you mean a series $f(x) = \sum_{n=0}^\infty \alpha_n x^n$ where $\alpha_{n+1}/\alpha_n$ is a rational function of $n$, then this is not closed under addition, subtraction, multiplication, division or composition. For example, take $f(x) = \sum_n x^n = 1/(1-x)$ and $g(x) = \sum_n 2^n x^n = 1/(1-2x)$ which are generalized hypergeometric, but $f+g$, $f-g$, $fg$, $f/g$, $g \circ f$ are not. They are closed under differentiation.
Robert Israel
- 448,999