THE RAYLEIGH FUNCTION NAND KISHORE

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1 THE RAYLEIGH FUNCTION NAND KISHORE The present paper is a study of a set of symmetric functions of the zeros of Jv(z), the Bessel function of the first kind. Let the zeros of z~'j,(z) be denoted by j,,m, m=l, 2,, where i?(>,m) g R(j,.m+i) And let (1) «T2.M - D (>.m)-2n, n = 1, 2,. m=i We propose to call a2 (v) the Rayleigh function of order 2«. The Rayleigh functions of odd orders are identically zero. These functions were first used by Euler to determine the three smallest zeros of Jo(2y/z); and Rayleigh, independently, calculated the smallest positive zero of Jv(z) with the aid of these functions. The Rayleigh functions have been the subject of a number of investigations by Cayley, Graeffe, Graf and Gubler, Watson, Kapteyn, Forsyth and others [6, p. 502]. In this paper we shall understand by the Bernoulli and Genocchi numbers, the entities Bn and Gn defined as follows: (2) *- W Bo = 1. B\ = 11 B2 = \, B3 = 0, Bi = -jpg-» (3) G = 2(1-2")Bn; (see [4; 3]). From the well-known formula VI z-1/2/i/2(z) = A/ sin z/z, (see [6, p. 54]), the roots of z~ll2jxß(z) are seen to be kir, k = l,2, Hence (j \ «> 22n 1 -) = ( T)-,n = x-2"f(2«) = (- I)""1 - B2n- 2 / *=i (2w)! Similarly, since zll2j-i/2(z) = \/2/ir cos z, the roots of zll2j-x/2(z) are Received by the editors December 14, 1961 and, in revised form, May 14,

2 528 NAND KISHORE [August (k i)tr, k=l, 2,. Therefore, <r (- \ = 7T-2» (k - -^ " = (22» - l)7r-2"f(2«) (5) 22» 2 (2re)! A generating function for a2niv) can be obtained. Let 7 (z) be expressed as a Weierstrassian product (r)' Mz) = r(* + i) n-l V J ; (see [6, p. 498]). Differentiating logarithmically with respect to z, CO Jliz)/J,(z) = v/z+y, i-2z)/ijl,k - z) i--»i = i/z{v-2±± n/ji:\, \ k=l n=l / CO zji (z)//,(z) = v - 2 ffh.m*»», 1 (6) za+i(z)/a(z) = cr2n(x)zs». 2 =,1 Substituting z= 1 in (6) yields (7) Í>*.(") =-A+1(D/A(l); n 1 2 (8) <rîs(». + *) - Jr+k+iil)/Jr+kil), * - 0, 1, 2,. n-l 2 Setting ï*= + and in (6), the following are obtained in view of (4) and (5), oo 22 (9) E (-l)"-1 T^TT 2 Z2» = 1 - z cot z, n-l (2«)! oo 22n~1 (10) Z (-l)n T-v G2»22" = 2 tan z. n=i (2 re)! Taking the continued fraction representation of the ratio of two

3 1963I THE RAYLEIGH FUNCTION 529 Bessel functions, /,+i(z)// (z) =-- 2(y+l)-2(v + 2)-, (see [6, p. 153)] then substituting z=l, and using (7), (H) E Cr2n(») = tí 2 2(v + 1) - 2(v + 2) - Setv= ±\ in (11), then " 22" 111 y (-i)-i-bu = (12) tí (2«)! = 1 - cot (1) =.3579, " 22" (13) Z(-1)" -G2 = T T -r n«i (2n)! = tan (1) = A recurrence formula for the functions a2n(v) may be derived from (6). Let (6), be written as 1 z/ +i(z) = J,(z) y.o-2n(v)z2n. 2 n=l Substituting the series for Jv(z) and / +i(z), and identifying the coefficients of z2n on both sides, A /n\/v + n\ (i4) z i- d^mtw ^ J ^ k j «r««=». Substitute v= + in (14), then in view of (4) and (5), A /2» + 1\ (15) 22*-i( )52*=W, *=i \ k / (see [5, p. 174]) " /2w\ (16) -S22*-2 ( )Ga =»; i=i \2 /

4 530 NAND KISHORE [August which are well-known Bernoulli and Genocchi recurrence formulas. A determinant representation of o2n(v) may be given. If in (14) the upper limit of the summation is taken as 1, 2, -, re, then re linear equations involving «functions o2(v), <ii(v),, o2n(v) are obtained. The determinant A of this system is triangular. Hence, the value of A is equal to the product of its elements on the main diagonal, A = (-l)»(«-i)/2.2''(»+1>-re!!n n 4=1 (" + k)n~k+l. Then by Cramer's rule, (17) o-2n(v) = (-1)""1- (»-1)11 4-» II» (" + Q-o+x-1 D, n\ k-i where, D = on «0 1 axthdcr)» 0 2 (re \ (v + n\ / re \ (v + re\ / re \ (v + n\ l)\ I ) \2)\ 2 )-\n-l)\n-l) " Substitution of v*= ± in (17) yields, respectively, (18) (19) re! A2 = 2-»+lDi, (2re + 1)! G2n = - 2-2»+27>2 where Di = 0 « /2re+lX /2re+lN % \ 2 / \2n-2j

5 1963] THE RAYLEIGH FUNCTION 531 >2 = (Î) (i)(i) n - (2n)» V 2 / \2n-2) Other formulas for a2niy) may be derived. Let (6) be written as {z/,+2(z)//,+i(z)hz/,+1(z)//,(z)} = 4 I E <r»(l> + l)z4 I cr2*«z2i} Then in view of the well-known formula, ) J,+2(z) =-J,+iiz) z - Jviz), (see [6, p. 45]), the above becomes X-Z2 + -i iv + l)z/,+1(z)//,(z) = { <r2*(v + l)z4 jí>2*mz4 4 2 (. i_i ; v t-i / Identifying the coefficients of z2n on both sides n-l (20) iv + l)cr2 W = cr2i(»< + l)<72n-2kiv). t=l Substitution of v=, in (20) yields «z*/2w\ (21) C?2n = ~ Z i ( -, ) B2kG2n-2kk=i \2k/ Consider the well-known (see [6, p. 45]), formulas d iz~"jriz)) = - z-"/ +i(z), dz It follows that - (z'+1/,+i(z)) = z>+v,iz). dz

6 532 NAND KISH0RE [August d_ /z"+lj,+i(z)\ A+l(2)\ dz\ z->j,(z) 'J,(z) ) ~ ît+1 (1 + A2+i(z) Jl(z) That is, (L I^X = _ +A2-,+11 (i. Zï^ z2a \2 A(z) / 4 2 da\2 7,(z) / Substituting (6) in this relation and identifying the coefficients of z2", n-l (22) iv + n)a-2niv) = Z o-uiv)o-2n-2kiv). k l Substitute v= ±$ in (22), then (23) - (1 + 2re) A2n = L, ) B2kB2n-2k, k=i \2k/ (see [5, p. 66]) ïz,1 /2re\ (24) -2(1-2re)G2n = (,. ) G2kG2n-2k. k-i \2k/ Let (6) be multiplied by itself, ll+iiz) = Alliz) ( o-2kiv)z2ka, then substituting the well-known series for J,+1iz) and J\iz), viz., s JL, r(2» k) / z \ 2"+2+2* ll+iiz) - Z(-l)*- I-( ) To *ir(2i» Ä)r*(»+2 + *)\2/ 2 - r(2y *) / z y+2* fo klti2v+l + k)t2iv+l + k)\2j (see [6, p. 147]), and identifying the following is obtained (25) the coefficients of z2n on both sides,» (2v + 2re - 2k\ /v + n\2 * *-i \ n k / \ k /,_i In view of (22) this reduces to /2k + 2re\ \ re- 1 /'

7 1963] THE RAYLEIGH FUNCTION 533» /2v + 2n - 2k\.. (v + n - 1\2 Eí-i)4-^ t ){( -i)!}2,, ) 0+ä) «rat«t_i Vf» */ Vä 1 / (26) _ 2k + 1 /2* + 2«- 2\ Substitute v= ±\ in (26), then 2v + «V» 1 / (27) (l + 2*)i f)5a= 2»-l, k=x \2k; VÍ (2n\ (28) (l-2*)( )Ga = 2(2»-l)GI,. *-i V2ä/ The Rayleigh functions of odd order are zero, «riw = 0, <r,f» = 0, «rsw = 0,. Identifying the coefficients of z2 in (6), an explicit expression for ff2(v) is obtained. Then (20) may be used to get explicit expressions for the Rayleigh functions of higher orders. Thus, o-2(v) = o-iiv) = o-eiv) = 0"Sw = 1 2\v + 1) 1 2«(v + l)*(v + 2) 2 2'(v + 1)3(" + 2)(v + 3) 5v+ 11 2«(F+l)«(i' + 2),(F + 3)(r + 4) The first twelve Rayleigh functions are given in [l, pp ]. Bibliography 1. D. H. Lehmer, Zeros of the Bessel function J,(x), Math. Comp. 1 ( ), , On the maxima and minima of Bernoulli polynomials, Amer. Math. Monthly 47 (1940), , Ann. of Math. (2) 36 (1935), E. Lucas, Theorie des nombres. 5. N. Nielsen, Traité élémentaire des nombres des Bernoulli. 6. G. N. Watson, A treatise on the theory of Bessel functions, Macmillan, New York, University of San Francisco