A generalization of the hyperbolic Pascal pyramid

In the present paper, we consider a variation of the hyperbolic Pascal pyramid where the three leg-sequences (the constant 1 sequence) are replaced by the sequences $$\lbrace \alpha _n\rbrace _{n\ge 0}$$ { α n } n ≥ 0 , $$\lbrace \beta _n\rbrace _{n\ge 0}$$ { β n } n ≥ 0 and $$\lbrace \gamma _n\rbrace _{n\ge 0}$$ { γ n } n ≥ 0 with $$\alpha _0=\beta _0=\gamma _0=\Omega $$ α 0 = β 0 = γ 0 = Ω , and we describe the values of elements. Then we give the recurrence relations associated to the sums of the values on levels in the generalized hyperbolic Pascal’s pyramids. The order of these recurrences is six.