  Recurrences for positive definite functions in terms of the space dimension
have been used in several fields of applications. Such recurrences typically
relate to properties of the system of special functions characterizing the
geometry of the underlying space. In the case of the sphere ${\mathbb S}^{d-1}
\subset {\mathbb R}^d$ the (strict) positive definiteness of the zonal function
$f(\cos \theta)$ is determined by the signs of the coefficients in the
expansion of $f$ in terms of the Gegenbauer polynomials $\{C^\lambda_n\}$, with
$\lambda=(d-2)/2$. Recent results show that classical differentiation and
integration applied to $f$ have positive definiteness preserving properties in
this context. However, in these results the space dimension changes in steps of
two. This paper develops operators for zonal functions on the sphere which
preserve (strict) positive definiteness while moving up and down in the ladder
of dimensions by steps of one. These fractional operators are constructed to
act appropriately on the Gegenbauer polynomials $\{C^\lambda_n\}$.
