Georgian Mathematical Journal Volume 15 (2008), Number 1, 1–20 FUNCTIONAL EQUATIONS AND µ-SPHERICAL FUNCTIONS MOHAMED AKKOUCHI, BELAID BOUIKHALENE, AND ELHOUCIEN ELQORACHI Abstract. We will study the properties of solutions f, {g i }, {h i }∈ C b (G) of the functional equation  G  K f (xtk · y) χ(k)dk  dμ(t)= n  i=1 g i (x)h i (y), x, y ∈ G, where G is a Hausdorff locally compact topological group, K a compact subgroup of morphisms of G, χ a character on K, and μ a K-invariant measure on G. This equation provides a common generalization of many functional equations (D’Alembert’s, Badora’s, Cauchy’s, Gajda’s, Stetkaer’s, Wilson’s equations) on groups. First we obtain the solutions of Badora’s equation [6] under the condition that (G, K) is a Gelfand pair. This result completes the one obtained in [6] and [11]. Then we point out some of the relations of the general equation to the matrix Badora functional equation and obtain explicit solution formulas of the equation in question for some particular cases. The results presented in this paper may be viewed as a continuation and a generalization of Stetkær’s, Badora’s, and the authors’ works. 2000 Mathematics Subject Classification: 39B42, 39B32, 43A90, 47G10. Key words and phrases: Locally compact group, functional equation, Gelfand pairs, Gelfand measure, K-spherical function, μ-spherical function. 1. Introduction Let G be a Hausdorff locally compact group and let K be a compact subgroup of morphisms of G. The action of k ∈ K on x ∈ G will be denoted by k ·x and the normalized Haar measure on K by dk. Furthermore, χ : K −→ {z ∈ C||z | =1} is a continuous homomorphism of K into the unit circle. Finally, we let µ denote a complex bounded measure on G. In this paper, we study functional equations of the form  G  K f (xtk · y) χ(k)dkdµ(t)= n  i=1 g i (x)h i (y), x, y ∈ G, (1.1) where f,g i ,h i are continuous and bounded unknown functions that we want to determine. The functional equation (1.1) is a generalization of many functional equations on groups. The functional equation (1.1) with µ = δ e : Dirac measure ISSN 1072-947X / $8.00 / c  Heldermann Verlag www.heldermann.de