## 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

### Keywords:

hypergeometric function

### Citations:

Zbl 1017.35081; Zbl 1029.35192; Zbl 0945.35063
Full Text:

### References:

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