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Generalisation of Jacobi's θ-Function Formulae


     

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Jacobi's well known formulae on the multiplication of θ-functions can be generalised by making use of general orthogonal linear substitution. The generalised theorem runs as follows:-

If variables (l, m, n,...... p) and (l1, m1, n1,..... p1) are connected by means of the relations

(i){a11l+a12m+........a1p P = a1ll1+a21m1+.......ap1 

      a2ll+a22m+ ...,...a2p P = a12l1+a22m1+......ap2P1

      ...........................................................................

      ............................................................................

     ap1l+ap2m+.........app P=a1pl1+a2pm1+......appP1

where aik=-aki.


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  • Generalisation of Jacobi's θ-Function Formulae

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Abstract


Jacobi's well known formulae on the multiplication of θ-functions can be generalised by making use of general orthogonal linear substitution. The generalised theorem runs as follows:-

If variables (l, m, n,...... p) and (l1, m1, n1,..... p1) are connected by means of the relations

(i){a11l+a12m+........a1p P = a1ll1+a21m1+.......ap1 

      a2ll+a22m+ ...,...a2p P = a12l1+a22m1+......ap2P1

      ...........................................................................

      ............................................................................

     ap1l+ap2m+.........app P=a1pl1+a2pm1+......appP1

where aik=-aki.