Orthogonal polynomial method and odd vertices in matrix mode(8)

We show how to use the method of orthogonal polynomials for integrating, in the planar approximation, the partition function of one-matrix models with a potential with even or odd vertices, or any combination of them.

ORTHOGONALPOLYNOMIALMETHODANDODDVERTICES9e2(g)oftheplanargraphs.ThiswillalsojustifythereplacementAj→A,Rj→Rusedintheprevioussection.Letusconsidertheidentity ′nhn=dµ(λ)λPn(λ)Pn(λ)

′=dµ(λ)Pn(λ)[Pn+1(λ)+RnPn 1(λ)+AnPn(λ)]

′=Rndλe S(λ)Pn(λ)Pn 1(λ)(26) =Rndλe S(λ)S′(λ)Pn(λ)Pn 1(λ)

=(1 k i=3i 1g¯iβn)hnRn,

whereinthelastbutoneequalitywehaveintegratedbypartsandiinthelastequalitywehaveusedthede nitionofβn.Thuswehaveobtainedthe rstrecursionrelation

n=(1 k i=3i 1g¯iβn)Rn.(27)

Fromthisequationweinferinparticularthat:Rn(0)=n.Wewantto ndasecondrecursionrelationwhichrelatesthecoe cientsAnandRn.Weobservethat:

′dλe S(λ)λPn(λ)Pndλe S(λ)Pn(λ)λS′(λ)Pn+1(λ)+1(λ)=

=(An+An+1 k i=3(28)ig¯iβn+1)hn+1.

But ′′dλe S(λ)λPn(λ)Pndλe S(λ)Pn 1(λ)Pn+1(λ)=nAnhn+Rn+1(λ)

=nAnhn+Rndλe S(λ)Pn 1(λ)S′(λ)Pn+1(λ)

=nAnhn hn+1Rn

Asaresult,thesecondrecursionrelationis

(An+An+1 k i=3ig¯iβn+1)Rn+1=nAn Rn+1Rnk i=3i 1g¯iγn.k i=3i 1.g¯iγn(29)(30)