Replacing the integer parameter in the family of chi square distributions by a continuous parameter leads to the family of the gamma distributions. A similar phenomena occurs with the non central chi square. What happens in higher dimensions, where the natural extension of the chi square family is the Wishart family? The extension to a continuous parameter is clarified by the Gyndikin theorem (1975), with a beautiful short proof by Shanbhag (1989). The case of the non central Wishart is really difficult, and its explicit solution has been conjectured by E. Mayerhoffer (2010). We prove this conjecture by a detailed analysis of the zonal polynomials of symmetric matrices and of the convolution of measures concentrated on singular semi positive matrices (Joint work with H. Massam).