On the Maximum of the Fundamental Functions of the Ultraspherical Polynomials

Special cases of this theorem have been proved by Erdös-Grünwald' and Webster2 (the cases a = 1/2 and a = 3/2) . If there is no danger of confusion we shall omit the upper index n in lk`, n ~ (x) . PROOF OF THE THEOREM . It clearly suffices to consider the lk(x) with -1 =< xk < 0 . From the differential equation of the ultraspherical polynomials' we obtain (") lk(xk) = {«~xk) _ axk z zP„ (xk) 1 xk Thus for xk < x <= xk+10 =< lk(x) <= 1 . Suppose now that k ~ 1, then we prove that in (xk_ 1 , Xk) lk(x) lies below its tangent at Xk . Denote by yl , y2 , • • • yn-1 the roots of lk(x) and by z1 , z2 , z„_2 the roots of lk' (x) . From (1) it follows that xk_1 < yk-1 < xk . To prove our assertion it suffices to show that zk1 > xk . xk = First we prove that yk_1 > xk-1 + 2 u. From (1)