Journal of Research of the National Bureau of Standards Vol. 48, No.2, February 1952 Research Paper 2294 Table of the Zeros and Weight Factors of the First Twenty Hermite Polynomials 1 Herbert E. Salzer, Ruth Zucker, and Ruth Capuano The chief use of this table of the zeros and weight factors of the Hermite polynomials is in the calculation of integrals over the interval [- co, co], when the integrand is either t he product of e-x' and a polynomial, or may be closely approximated by ex' times a poly­ nomial. The zeros and weight factors, xln) and a ~n), respectively, together with the aux­ iliary quantities ,Bln)=a~n) exp [(xln»)2 ], which are useful in computation, are all tabulated here for t he first twenty H ermite polynomials. T he zeros xln) and ,B~n) are given to 13 or more decimals, and t he weight factors a/ n ) to 13 significan t figures. Although other shorter t ables have appeared , this present table will enable one to cope with problems requiring much higher degree and accuracy, both ir. problems involving direct quadratures and those arising in t he numerical solu tion of linear integral equations for the range [- CX), co ]. Thus t he use of this t able in any direct quadrat ure can give exact accuracy as far as the 39th degree (the only inexactitude arising from the use of rounded values of x ~n) and a\n»). 1. Introduction By considering ex2 l (x) R (y, x) a a polynomial in x, and making use of (2) for y =Xln), i= 1, 2, ... , n, The main purpose of the table of the zeros Xln ) the approximate olution of (3) is reduced to the and weight factor (" Christoffel numbers") a ln ) of solution of a set of only n linear equation for the the HermiLe polynomial is to provide the basic (approximate values of) l (x\n»). These alues of quantities that occur in th e quadrature formula f (x;"») can then be used in (2) to find f ey ) for any value of y. The advantage in the use of (2) instead (1) of an equally spaced quadrature formula for olving (3) is that here only n points are needed to give where accuracy obtainable by approximating ex2 f (x) R (y,x) 7r !j<2n) W as a polynomial of the (2n-1) th degree in x. For a full description of the method, including examples, Rn = 2 n (2n) (2n- 1) ... (n+ 2)(n+ 1) see A. Reiz [1], especially p. 4 to 10, 16 to 21. for some ~, - (Xl < ~ < (Xl [2 , p. 101- 102,369].2 When the integrand does not contain e-x2 explicitly, 3 . Important Properties of Hermite but behaves like e-x2 times a polynomial, instead of (1), there is Polynomials The Hermite polynomials may be defined by the formula For dealing with quaclratures of the type in (2), it (4) i convenient to follow the example of A. Reiz [1] and to tabulate also the quantities (3 \n)= a ln) exp [(x\n»)2 ]. From the form of the remainder term R n , or by the generating function (1) is seen to be exact (save for the use of rounded values of x\n) and a \n») when l ex) is a polynomial of at most the (2n- 1) th degree. (5) 2 . Numerical Solution of Integral Equations They are given explicitly by the formula Be ides problems involving direct quaclratures, ~ (- l)kn ! there are those arising in the numerical solution of H n(x) = L.; k'( - 2k)' (2X)n-2k. (6) linear integral equations of the second kind for k=O • n . integrals over the range [- (Xl, (Xl ], namely, The numerical values of the coefficients of H n(x) for the first 30 values of n are contained in H. E . Salzer cf> 0;)= f (Y)+A f :c/ (x)K (y, x) dx. (3) [3]. The polynomials H n(x) atisfy the differential ----- equation I This project was sponsored in part by the Omce of Air Rcsearch, AMC, USAF. 2 Figures in brackets indicate the literature references at the end of this paper. H n" (x) - 2xHn'(x) + 2nHn(x) = 0, (7) III the 'recurrence formula n~2 , (8) and the differential-recurrence formula by taking twice the square of the latter. The poly­ nOJIlials A rn/21~ X ) and B rn /2 ]( X ) were t abulated for five or six values of X in the neighborhood of each H n' (x) = 2nH n _ 1 (x). (9) X,c n) , at intervals of 0.0001 in X . The zeros x ,c") The Hermite polynomials have the orthogonality were calculated by a formula for inverse interpolation property given in H . E. Salzer [6], and then were checked by direct interpolation in the polynomials A rn /21( X) and B rn /21( X ), using the Gregory-Newton formula with advancing differences. The accuracy of the zero X i(") was determined from the approximate relation: A most important consequence of (1 0) is (1), where n error in X Cn) !:!!. 0,0001 F(X.(n») the al ) are defined by ,- t:. " (11) where F (X,<n») denotes either A rn /21( X ,cn») or B rn /2] (X ,c"»), and t:. is the difference between two consecutive entries of F (X) on either side of X iCn). The a \n) are also expressible in the following form The values of xln) were found from x,cn) = [2, p. 344], [4, p. 311]: 1 - ± 2..jx,cn) , and checked back by squaring to get ?T ! 2 n+1n! a Cn ) - ~=-o-~--,-,-.-;: (12) (n ) 2. i -( H n' (xl"»)P' X ln) = 4xi Additional checks upon the xi''' were performed by ealculating which, from (9), can be written as ±xt)2=?1'(n-l), (14) i=l 2 (13) ;'x.(n)=(_1)<n-IJ/2 (n+ 1)! for n odd (15a) the latter form being more convenient for compu­ i=l ' ((n + 1)/2)! ' tation. where ?T' denotes the omission of the factor xin ) = 0, 4 . Description of Tables and , This table gives the zeros xiC,,) , the weight factors n n' ?T x i(n)=(- 1)n/2_-' - for n even. (15b) aln ' and (3i(n), for the first 20 Hermite polynomials. i=l (n/2)! The absolute values of the zeros, Ixl") I, are tabulated to 15 decimals for n = 1(1)13, to 14 decimals for The weight factors a i(nJ were calculated from'" (13 ), n=14(1 )16 , and to 13 decimals for n = 17 (1)20, To using - each nonvanishing Ixi(") I there corresponds the two zeros xi(n) = + Ixln)I and xiCn ) = -Ix,"') I. The weight factors a,Cn) = a~n~i +1 are tabulated to 13 significant figures. For a given n, zeros having the same ab­ and solute value are associated with the same weight factor. The auxiliary quantities (3,c") are tabulated to 13 decimals. Formula (12) was used to check the ai(n), in the form 5. Method of Computation and Checking ?T ; 2 n - 3n ! . 1 Since H n(x) is an odd or even polynomial for n odd { X ,"') A~n /21 (X r») } 2' for n odd, ~ ~2 (n+ 1) (12a) or even, respectively, it was convenient to change the variable to X = 4X2 and to calculate X !n) = 4x.Cn )2, and where the nonvanishing X,c" ) are the zer~ s of p ~ l y­ a,(n) X !n) {B ' (X .Cn») }2' for n even . (12b) nomials A rn/2l (X), n · odd, and B rn /2 ](X), n even, of , (n/21, degree [n/2] in X. Thus the degree of the polyno­ mials A (n/2](X) and B rn /21(X) did not exceed 10, and the coefficients were considerably smaller than those Additional over-all checks upon a ,<n) were performed, of H n(x). A first approximation to Xi(n) was ob­ using (1) for P (x) = l , X2 , x\ ... , x2m, 2m < 2n- 1< tained from E . R. Smith [5], which gives the zeros of 2m+\ and in a few cases for P (x)=x2n+2P, 2m+ 2P< 112 pens that that integral is lrnown to be ~~ 7r e- I/ 8I o(1'; ' 2n- l < 2m+l , the left member of (1) equal tor (k +~) where I o(t) denotes the modified Be el function of for P (X)=X2k• The choice of these different powers order zero Csee [26, p. 394]) . As the integrand of x for P (x) wa to en ure that every significant e- 12J O(t) i even, and as ~(") Jo(x,(n»)= a~n2'+1 figure of the at.) was covercd in the checks. J o (- x?» ), from (1), whenever n is even, The auxiliary quanitites (3,(") were computed by multiplying the weight factors ain) by exin ) /4, where i a> e-12 J o( t) dt~~aIn) J o(x,<"»), (16) the ex,(n)/4 were obtained from the NBS tables of the exponential function [7] and were checked by the where the summation covers only the positive value n n relation log (3 i" ) = log ai ) + X i ) /4, where the loga­ of x/n ). In the present example the error in (16) rithms were obtained from the NBS tables [8] . (from the term Rn in (1) and the inequality The functions x,(n) , ar), and (3,(n) were calculated !J Jm) (x) ! ~ 1) does not exceed 10 - 13 for n as small as 10. to about two more places than are tabulated, and For the right member of (16 ), J o(xPO») was computed are available in manuscript form.