Orthogonal polynomial method and odd vertices in matrix mode(7)

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.

8ORTHOGONALPOLYNOMIALMETHODANDODDVERTICESofistepswhose nale ectistogouponestepis

[i+1

(i 2p+1)!p!(p 1)!,(19)

ithecontinuumwhere[]standsfortheintegerpart,and,denotingbyβivalueofβn,wehave

[i+1

i=β(i 2p+1)!p!(p 1)!Ai 2p+1Rp 1(20)

wherethetilderemindsthereplacementAj→A,Rj→R.Thevalues iforthe rstfewiareofβ

2=2Aβ

3=3A2+3Rβ

4=4A3+12AR.β

Analogouslywede ne

ihn+1γn=(21) dµ(λ)Pn+1(λ)λiPn 1(λ).(22)

iBythesametechniqueusedforβnwe nd

[i

γ i=

Finallywede ne(i 2p+2)!p!(p 2)!Ai 2p+2Rp 2.(23)

[i+1

i=β i Rγ i 1=δ i 1 Aβ(i 2p+1)!(p 1)!2Ai 2p+1Rp 1.

(24)

iforthe rstfewiareThevaluesofδ

3=A2+2Rδ

4=A3+6ARδ

5=A4+12A2R+6R2.δ

4.Derivationofthecontinuumequations

InthissectionweshallexaminethecontinuumlimitN→∞,whichwillallowustowriteasimpleexpressionforthegeneratingfunction(25)