Is there any closed form expression for the series $1+\displaystyle\frac{(1/2)}{1!}\frac{z}{1!}+\frac{(1/2)(3/2)}{2!}\frac{z^2}{2!}+\frac{(1/2)(3/2)(5/2)}{3!}\frac{z^3}{3!}+\cdots$? I tried with exponential series like taking $e^{z/2}$. It didn't work. Can someone give me some hints?
2026-03-25 00:02:18.1774396938
Closed form the following series
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This is the hypergeometric function $_{1}F_{1}(1/2, 1, x).$ Expressed with the modified Bessel function of the first kind $I_0$ your series is
$$f(x) = e^{x/2}I_0(\tfrac{1}{2} x) = 1+\frac{1}{2} x+\frac{3}{16}x^2+\frac{5}{96}x^3+\frac{35}{3072}x^4+\frac{21}{10240}x^5+O(x^6)$$