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Fundamental solutions of the Tricomi operator. III. (English) Zbl 1083.35068

In this paper the authors complete their results with respect to the Tricomi operator: \[ \operatorname{Im} =y\;\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}\,. \] Let \((\xi,\eta)\) be a point in \(\mathbb R^2\). A distribution \(K_{(\xi,\eta)}(x,y)\in {\mathcal D}'(\mathbb R^2)\) is said to be a fundamental solution of \(\operatorname{Im}\) relative to \((\xi,\eta)\) if \[ \operatorname{Im}_{x,y}K_{(\xi,\eta)}(x,y) = \delta_{(\xi,\eta)}(x,y), \] where \(\delta_{(\xi,\eta)}(x,y)\) is the Dirac distribution at \((\xi,\eta)\).
Explicit expressions for fundamental solutions of \(\operatorname{Im}\) relative to any point in the elliptic, parabolic, or hyperbolic region of it are obtained. It is revealed the important role that the hypergeometric function \(F(1/6, 1/6; 1;\zeta)\) plays in finding such solutions. In view of the invariance of \(\operatorname{Im}\) under translations along the \(x\)-axis and in order to simplify notation, it suffices to consider the case when \((\xi,\eta)=(0,b)\) is an arbitrary point along the \(y\)-axis. There are then three cases to be dealt with, namely, \(b>0\), \(b=0\), and \(b<0\). Some basic facts on hypergeometric functions are reviewed and various technical results are proved.
For parts I and II see ibid. 98, No. 3, 465–483 (1999; Zbl 0945.35063), 111, No. 3, 561–584 (2002; Zbl 1017.35081) and 117, No. 2, 385–387 (2003; Zbl 1029.35192).

MSC:

35M10 PDEs of mixed type
45F05 Systems of nonsingular linear integral equations
35A08 Fundamental solutions to PDEs
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