Hypergeometric series ${}_3F_2$

Question

Is there any general formula for ${}_3F_2[a,b,c;2b,2c;z]$? Kindly share any textbook or any research paper on this type of hypergeometric series.

Answer 1

Kindly share any textbook or any research paper
For a large collection of formulas, see functions.wolfram.com

Wolfram Alpha does not know a formula for the case $a=1,b=2,c=2/3$.

Answer 2

Some Research Paper would be Math World and Wolfram Functions.

Answer

Such a general series does not exist, since $2 \cdot b$ or $2 \cdot c$ could be non-positive integers, which, according to the definition of the generalized hypergeometric function, involves taking the gamma function of a non-positive integer or dividing by $0$, but that's not defined, so the series is undefined aka does not exist.

Ther general series for $\left\{ -b,\, -c \right\} \not\in \mathbb{N_{0}}$ would be: $$ \begin{align*} \operatorname{_{3}F_{2}}\left[ a,\, b,\, c;\, 2 \cdot b,\, 2 \cdot c;\, z \right] &= \sum\limits_{k = 0}^{\infty}\left[ \frac{\Gamma\left( 2 \cdot b \right) \cdot \Gamma\left( 2 \cdot c \right) \cdot \Gamma\left( a + k \right) \cdot \Gamma\left( b + k \right) \cdot \Gamma\left( c + k \right)}{\Gamma\left( 2 \cdot b + k \right) \cdot \Gamma\left( 2 \cdot c + k \right) \cdot \Gamma\left( a \right) \cdot \Gamma\left( b \right) \cdot \Gamma\left( c \right) \cdot k!} \cdot z^{k} \right]\\ \end{align*} $$

Apart from that, there are still convergence conditions for the individual components as you can see here.