We consider the family P of n-tuples P consisting of polynomials P1,...,Pn with nonnegative coefficients, which satisfy partial derivative iPj(0)=delta i,j,i,j=1,...,n. With any such P, we associate a Reinhardt domain triangle P<mml:mspace width="-0.166667em"></mml:mspace>n</mml:msubsup> that we call the generalized Hartogs triangle. We are particularly interested in the choices Pa=(P1,a,...,Pn,a),a0, where Pj,a(z)=zj+a Pi k=1n</mml:msubsup>zk,<mml:mspace width="3.33333pt"></mml:mspace>j=1,...,n. The generalized Hartogs triangle associated with Pa is given by: <disp-formula id="Equ84"><mml:mtable><mml:mtr><mml:mtd columnalign="right">triangle a<mml:mspace width="-0.166667em"></mml:mspace>n</mml:msubsup>=</mml:mtd><mml:mtd columnalign="left">{z is an element of CxCn-1</mml:msubsup>:|zj|2<|zj+1|2(1-a|z1|2),<mml:mspace width="3.33333pt"></mml:mspace>j=1,...,n-1,</mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="right"></mml:mtd><mml:mtd columnalign="left"><mml:mspace width="1em"></mml:mspace>|zn|2+a|z1|2<1}.</mml:mtd></mml:mtr></mml:mtable><graphic position="anchor" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="40687_2023_397_Article_Equ84.gif"></graphic></disp-formula>The domain triangle 0<mml:mspace width="-0.166667em"></mml:mspace>2</mml:msubsup> is the Hartogs triangle. Unlike most domains relevant to the multi-variable operator theory, the domain triangle P<mml:mspace width="-0.166667em"></mml:mspace>n</mml:msubsup>,n2, is never polynomially convex. However, triangle P<mml:mspace width="-0.166667em"></mml:mspace>n</mml:msubsup> is always holomorphically convex. With any P is an element of P and m is an element of Nn, we associate a positive semi-definite kernel KP,m on triangle P<mml:mspace width="-0.166667em"></mml:mspace>n</mml:msubsup>. This, combined with the Moore's theorem, yields a reproducing kernel Hilbert space Hm2</mml:msubsup>(triangle P<mml:mspace width="-0.166667em"></mml:mspace>n</mml:msubsup>) of holomorphic functions on triangle P<mml:mspace width="-0.166667em"></mml:mspace>n</mml:msubsup>. We study the space Hm2(triangle <mml:msub>P<mml:mspace width="-0.166667em"></mml:mspace>n) and the multiplication n-tuple <mml:msub>Mz acting on Hm2(triangle <mml:msub>P<mml:mspace width="-0.166667em"></mml:mspace>n). It turns out that <mml:msub>Mz is never rationally cyclic, but Hm2(triangle <mml:msub>P<mml:mspace width="-0.166667em"></mml:mspace>n) admits an orthonormal basis consisting of rational functions on triangle <mml:msub>P<mml:mspace width="-0.166667em"></mml:mspace>n. Although the dimension of the joint kernel of Mz-lambda is constant of value 1 for every lambda is an element of triangle <mml:msub>P<mml:mspace width="-0.166667em"></mml:mspace>n, it has jump discontinuity at the serious singularity 0 of the boundary of triangle <mml:msub>P<mml:mspace width="-0.166667em"></mml:mspace>n with the value equal to infinity. We capitalize on the notion of joint subnormality to define a Hardy space H2(triangle <mml:msub>0<mml:mspace width="-0.166667em"></mml:mspace>n) on the n-dimensional Hartogs triangle triangle <mml:msub>0<mml:mspace width="-0.166667em"></mml:mspace>n. This in turn gives an analog of the von Neumann's inequality for triangle <mml:msub>0<mml:mspace width="-0.166667em"></mml:mspace>n.
