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On Hypergeometric Series and OEIS Sequence

Mathematics Asked by Mr. N on December 5, 2021

I have been searching an integer sequence in OEIS. The sequence is the following: OEIS A321234 (https://oeis.org/A321234) . So far, so good. However, this sequence is the denominator of a Hypergeometric Series, the following one:

$${}_3 F_2([1/2, 1, 1], [3/2, 3/2], x).$$

The problem is: I do not even know what those kind of series are. Do not even know what this notation means. Could someone recomend me any book references so as to understand it better? I have read Wiki’s page, but it seems not enough.

Thanks a lot

One Answer

$$, _3F_2left(frac{1}{2},1,1;frac{3}{2},frac{3}{2};xright)$$ is one of the many hypergeometric functions (google for that).

They are very special functions corresponding to infinite sums. For this one, the first terms of its expansion are $$1+frac{2 x}{9}+frac{8 x^2}{75}+frac{16 x^3}{245}+frac{128 x^4}{2835}+frac{256 x^5}{7623}+frac{1024 x^6}{39039}+frac{2048 x^7}{96525}+frac{32768 x^8}{1859715}+Oleft(x^{9}right)$$

You would find the numerators in sequence $A046161$

Answered by Claude Leibovici on December 5, 2021

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