mathematics of computation, volume 24, number 111, july, 1970 The Calculation of Multidimensional Hermite Polynomials and Gram-Charlier Coefficients* By S. Berkowitz and F. J. Garner Abstract. The paper documents derivations of: (a) a recurrence relation for calculating values of multidimensional Hermite polynomials, (b) a recurrence relation for calculating an approximation to the Gram-Charlier co- efficients of the probability density distribution associated with a random process, based on (a), (c) an efficient algorithm to utilize the formulae in (a) and (b). I. Introduction. The paper documents derivations of: (a) a recurrence relation for calculating values of multidimensional Hermite polynomials; (b) a recurrence relation for calculating an approximation to the Gram-Charlier coefficients of the probability density distribution associated with a random process, based on (a); (c) an efficient algorithm to utilize the formulae in (a) and (b). The relations for producing Hermite polynomials are generalizations of formulae derived in Appell and Kampé de Fériet [1]. The recurrence relation mentioned in (a) above appears without derivation in Vilenkin [5], which further references a disserta- tion by Sirazhdinov [4]. Since Hermite polynomials form complete, biorthogonal systems with respect to the Gaussian probability density, one basic use of the polynomials is to expand a near-Gaussian probability density distribution in terms of the polynomials in a so- called Gram-Charlier series. As we shall demonstrate below, the coefficients of the series can be calculated directly from the time series generated by a random process. Finally, we present and prove an algorithm which computes a Hermite polynomial or Gram-Charlier coefficient of vector order m by means of the above recurrence relations. The algorithm requires the smallest number of suborder polynomials and/or coefficients possible. The derivation of the recurrence relations is the responsibility of S. Berkowitz, and the algorithm represents the work of both authors. The algorithm has been im- plemented in FORTRAN subroutines by Mr. Garner. II. Multidimensional Hermite Polynomials. A. Definitions. 1. m = (m,, • ■• , m„) is an order vector of degrees m¿. The polynomial or argu- ment x and of index m is said to be of degree m, in the argument xt (/ = 1, •••,«) and of total degree m = XX i m<- Received August 22, 1969. AMS Subject Classifications. Primary 3340; Secondary 6020, 6525. Key Words and Phrases. Multidimensional Hermite polynomials, Gram-Charlier series, recurrence relations, Gaussian probability density, normal distributions. * This paper was written in the Applied Mathematics Laboratory of the Naval Ship Research and Development Center for Project No. 1-830-918-02. Copyright © 1971, American Mathematical Society 537 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 538 S. BERKOWITZ AND F. J. GARNER 2. x = (x,, • • • , xn) is a vector of arguments x,. 3. R = Q~l is a positive definite matrix. One calls R a covariance matrix for the particular application of expanding joint probability density distributions in a Gram- Charlier series. R =- [r,,]; Q m \qit\. 4. The complete biorthogonal systems of Hermite polynomials {//m(x)| and |Gm(x)¡ are defined by the generating functions: (J) exp [sYQx - WO*-) = ¿ cm(a)ffm(x), OS (2) exp [aTx - ia**a] = E cm(a)Gm(x), »-0 where (O cm(a>= n:-i «CMl, (ii) the a, are arbitrary. The sets {77m(x)| and |Gm(x)S are biorthogonal with respect to (det Ö)1/2(2x) "'2 •exp [—§xrgx] over the entire real «-space. In the case of unidimensional poly- nomials (« = 1), both \Hmix)\ and {C7m(x)¡ reduce to a single orthogonal, complete set {Hjix)}, which is generated by the following relation (cf. Erdélyi [2]): » m (3) exp [ax - \a2] = Y3,£ —ml Hm(x). The polynomials {Hm(x)\ are orthogonal with respect to exp [—§x2] over the entire real line. B. Recurrence Relations for Hmix) and Gm(x). Differentiating the generating functions (1) with respect to ak produces ¿i n cc — ¡exp [arÖx - a7£?a]¡ = X <?*,(*, - a,) 2~2 cm(a)//m(x) (4j oak (_, „,,0 - z(n^), flr'ni g-w. ¿=o \irt rnAI (mt — 1)! By matching coefficients of a™*"1U^t af' in the right-hand equation of (4) one finds Hmix) - ( X) <?*,*;Jtfm-etto — £ it, «', ^m-ei-e, (X) _ ?«(»"* ~ D^m^U), ,<r-) * /-I / ,Vt * = 1, ••• , n, where et is a vector with a "1" as the /cth component, and zeroes elsewhere. A similar operation on the generating function (2) produces: Gm(x) = xtGm-tki±) — Jl rk¡mjGm-tt-t¡(x) — imk — l>tiGm_2et(x), (6) "' , . k = 1, • ■• , n. For any particular //m(x) or Gm(x), one has a choice of « recurrence relations. The value of any term in (5) or (6) with a negative index is zero and the jth relation must not be used if m, = 0 as one would falsely conclude Hm(x) = 0. If one chooses the pXh recurrence relation to compute Hmix) or Gm(x), then mv is called the pivot order, and p is called the pivot. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use HERMITE POLYNOMIALSAND GRAM-CHARLIERCOEFFICIENTS 539 III. Calculation of Gram-Charlier Series Coefficients. Consider the «-dimen- sional vector time series x<u, x<2), ... generated from a random process whose joint probability density is pix). The components of the kth vector of the time series are: (7) x<w=*i\— ,x?>. Denote the Gaussian probability density by the formula: p0{x) = (det 0),/2(27r)-/2 exp [-±(x - u)T0(x - »)], where (a) y = Pi , ■■ • , m»is usually taken as the vector of ensemble averages of the variâtes xx , • ■• , x„ , respectively. (b) Q is a positive definite n by n matrix, usually taken as the inverse covariance matrix of x. One can expand pix) in a Gram-Charlier series as follows: co pix) = p,fx) XI AmH,Jy), where (a) y = yu • • • , yn is a normalization of x such that y¡ = (x< — p,) <ru (/ = 1, • • • , «), and (b) a, = £[(x, — j»,)2] is the second moment of x, about its expectation. The biorthogonality property of the 7/m(x) and Gm(x) is: (8) / p„ix)Gmix)Hp(x) dx = bm H 8-iPi. J s ¿-1 where: (a) S indicates the entire real «-space, (b) bm= n:.i wr\ (c) 5mp is the Kronecker delta. By means of the biorthogonality property (8) and the Gaussian weighting function Paix), one can formulate the Gram-Charlier coefficients as follows: Am = bm / Gm(y)p(x) dx, J s or, simply, (9) Am = bmE[Gmiy)]. From (9) and the recurrence relation (6), one deduces : (10) Am = bm^E[ykGm.eiiy)] - ¿ r„m<£[Gm_.è_.,(y)] + rlt£[Gm 2et(y)]J. For a sample set of M vectors of the type indicated by (7), the first term on the right-hand side of (10) may be written as 1 M (ID £Lv*Gm_ei(y)]« — 2Z yi°Gm^t(Y°). M ,«,, License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 540 S. BERKOWITZ AND F. J. GARNER The expectations pk and <rhhave approximations i M (12) p»«t;£*"\ (13) ^«¿Ztó" "M2. Thus, by substituting (11) and (9) into (10) one has a recurrence relation for A^ (which, in turn, employs the recurrence relation (6) for Gm(x)) : (14) Am = jrz [ôm £ ^i)Gm_ei(y<") - ¿ rtimY Am.et.etJ , k - 1. ■ ri. Similarly, for the expansion eo /?(x) = p0(x) X BmGm(y), one can derive the recurrence relation: j I- JT * n "I m« M L i-x i-x i-i J k = 1, • ■• , «. As in the recurrence relations for Hermite polynomials, any term in (14), or (15) with a negative index is to be regarded as zero. In fact, since the indexing in (14) and (15) corresponds (with a bit of manipulation) to the indexing in (5) and (6), the problems of sequencing the recursive generation of Gram-Charlier coefficients are the same as those for sequencing Hermite polynomials, with the exception that only the vectors m — ek — eh f = I, ■■ ■ , n, need be generated for the Gram-Charlier coefficients. Therefore, in the remainder of the paper, we restrict ourselves to a discussion of Hermite polynomials and make only parenthetical restrictions where necessary for Gram-Charlier coefficients. IV. An Algorithm for Generating Values of Hermite Polynomials (or Gram- Charlier Coefficients). In recursively generating the value of Hmix) or Gm(x) from (5) or (6) one could conceivably generate all i7,(x) or Gv(x) such that v, g mt(i= 1, • • ■ , n). However, there is a test that permits one to determine, purely on the basis of the order vector v, whether or not a polynomial value need be calculated from the recurrence relation. The test is embodied in an algorithm which we present below and prove in the next section. The algorithm requires a prior decision on the generating sequence as follows: The order vector m is permuted according to an arbitrary but fixed pivot sequence vector (16) 6 = o-(l), ••• ,<r(«), where the set {o-(y')j is a permutation of 1, • • • , «. The classes Vv, defined by License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use HERMITE POLYNOMIALS AND GRAM-CHARLIER COEFFICIENTS 541 Vv = {v = v¡, • ■■ ,vn | 0 g vrii) = mcU), 1 ú i Ú P — 1; 0 < »,(,) Ú m,M;u,u) =0,p+lè/^«), p = 1, •• • , n, are each generated according to an ordering Sv which is defined by Sv = •Sv.v' E ^p | v = (o,(1), • • • , D,(P), 0, • • • , 0), v' = (oi(I), - • * . »í(p>» 0, • • • ,0), and v < v' iff X ("»<» — fíen)**-1 < 0, where b > max imt)f.