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Recurrences for positive definite functions in terms of the space dimensionhave been used in several fields of applications. Such recurrences typicallyrelate to properties of the system of special functions characterizing thegeometry 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 theexpansion of $f$ in terms of the Gegenbauer polynomials $\{C^\lambda_n\}$, with$\lambda=(d-2)/2$. Recent results show that classical differentiation andintegration applied to $f$ have positive definiteness preserving properties inthis context. However, in these results the space dimension changes in steps oftwo. This paper develops operators for zonal functions on the sphere whichpreserve (strict) positive definiteness while moving up and down in the ladderof dimensions by steps of one. These fractional operators are constructed toact appropriately on the Gegenbauer polynomials $\{C^\lambda_n\}$.

One-Step Recurrences for Stationary Random Fields on the Sphere