Communications of the
Korean Mathematical Society CKMS

The main objective of this paper is to establish certain explicit expressions for the Humbert functions $$ \Phi_2 (a,\,a+i\,;\,c\,;\,x,\,-x) \quad \text{and} \quad \Psi_2 (a\,;\, c,\,c+i\,;\,x,\,-x)$$ for $i=0,\, \pm 1,\, \pm 2,\, \ldots,\, \pm 5$. Several new and known summation formulas for $\Phi_2$ and $\Psi_2$ are considered as special cases of our main identities.

Keywords: gamma function, Pochhammer symbol, hypergeometric function, generalized hypergeometric function, Kummer's second summation theorem, Humbert's double double hypergeometric functions

MSC numbers: Primary 33C15, 33C70; Secondary 33B15, 33C05