Integral Representations of Ultraspherical Polynomials II
SSubmitted exclusively to the London Mathematical Society doi:10.1112/0000/000000
Integral Representations of Ultraspherical Polynomials II
N. H. Bingham and Tasmin L. Symons
In memory of Dick Askey, 4th June 1933 – 9th October 2019
Abstract
In the first part, by the first author’s work of 1972, an integral representation for an ultrasphericalpolynomial of higher index in terms of one of lower index and an infinite series was obtained.While this representation works well from a theoretical point of view, it is not numericallysatisfactory as it involves polynomials of high degree, which are numerically unstable. Here wesum this series to obtain an integral, which is numerically tractable. Introduction
As in [ ], we write W λn ( x ) for the ultraspherical (or Gegenbauer) polynomial of degree n and index λ > , p. 302], [ , § W λn (1) = 1. This choiceof normalisation, due to Bochner [ ], is convenient probabilistically; the first part [ ] wasprobabilistically motivated [ ]; so too is this sequel (see § W νn are orthogonal polynomialson the interval [ − ,
1] with respect to the probability measure G ν , where G ν ( dy ) = Γ( ν + 1) √ π Γ( ν + 1 /
2) (1 − y ) ν − / dy : (1.1) (cid:90) − W νm ( x ) W νn ( x ) G ν ( dx ) = δ mn ω νn , where ω νm = n + νν Γ( n + 2 ν ) n !Γ(2 ν ) . (1.2) Theorem 1.1 [ ]. If < ν < λ , x ∈ [ − , , there exists a probability measure M λν ( x ; dy ) on [ − , such that W λn ( x ) = (cid:90) − W νn ( y ) M λν ( x ; dy ) . (1.3) Moreover, when λ (cid:54) = ν the measure M λν is absolutely continuous with density M λν ( x ; dy ) = G ν ( dy ) ∞ (cid:88) m =0 ω νm W λm ( x ) W νm ( y ) . (1.4)The series in 1.4 is transparent from a theoretical point of view (it is derived in [ ] from theearlier work of Askey and Fitch [ ] by an Abel-limit operation), but unsuitable for numerical Mathematics Subject Classification a r X i v : . [ m a t h . C A ] J a n age 2 of 7 N. H. BINGHAM AND TASMIN L. SYMONS use as it involves polynomials of high degree, which oscillate wildly. Our purpose here is tocircumvent this by giving an explicit formula for the sum of the infinite series as a doubleintegral , which is numerically tractable. Our result, Theorem 3.1 below, is interesting in itsown right, completing the integral representations in [ ] by showing the dependence on thehigher index, λ , in a more convenient and structurally revealing way.2. Preliminaries
The Poisson kernel for the Jacobi polynomials reduces in the ultraspherical case to thegenerating function ∞ (cid:88) n =0 ω νn r n W νn ( x ) = (1 − r ) / (1 − rx + x ) ν +1 , r ∈ ( − , , (2.1)cf. [ , (2.1)]. Note that this is not the usual generating function for the ultrasphericalpolynomials [ , § ] showed that for x, y ∈ [ − , r ∈ ( − , ≤ ν < λ ≤ ∞ , the series ∞ (cid:88) n =0 ω νn r n W λn ( x ) W νn ( y ) (2.2)converges to a non-negative sum-function, which leads to a corresponding probability measure M λν ( x ) satisfying W λn ( x ) = (cid:90) − W νn ( y ) M λν ( x ; dy ) , n = 0 , , , . . . . (2.3)Here (see [ ]) we may take 0 ≤ ν ≤ λ ≤ ∞ , x ∈ [ − , x = ± M λν ( ±
1) = δ ± (as W λn ( ±
1) = ( ± n ). If λ = ν , then M λλ ( x ) = δ x (as there is no projectionto be done); so we may restrict to ν < λ as before. Now [ , Lemma 1] gives the Abel-limitoperation explicitly: for x, y ∈ ( − , r = 1 here to get m λν ( x ; y ) := ∞ (cid:88) n =0 ω νn W λn ( x ) W νn ( y ) ≥ , (2.4)a non-negative function in L ( G ν ), finite-valued unless x = y and ν < λ ≤ ν + 1. It is in factthe Radon-Nikodym derivative dM λν ( x ; dy ) /dG ν ( dy ): M λν ( x ; dy ) = G ν ( dy ) · m λν ( x ; y ) = G ν ( dx ) · ∞ (cid:88) n =0 ω νn W λn ( x ) W νn ( y ) . (2.5)Following [ ], for λ > ν write H λν for the probability measure of Beta type on [0 ,
1] given bythe
Sonine law H λν ( dx ) := 2Γ( λ + )Γ( ν + )Γ( λ − ν ) · x ν (1 − x ) λ − ν − dx. (2.6)This occurs in Sonine’s first finite integral for the Bessel function [ , p. 373]: forΛ µ ( t ) := Γ( ν + 1) J ν ( t )( t/ − µ , (2.7)Λ λ − ( t ) = (cid:90) Λ ν − ( ut ) H λν ( du ) (2.8)(the drop by a half-integer in parameter here reflects the drop in dimension in S d ⊂ R d +1 ; see § EPRESENTATIONS OF ULTRASPHERICAL POLYNOMIALS
Page 3 of 7For the product of W n terms in (2.4), we need Gegenbauer’s multiplication theorem for theultraspherical polynomials [ , p. 369], W νn ( x ) W νn ( y ) = (cid:90) − W νn ( xy + σ (cid:112) − x (cid:112) − y ) G ν − ( dσ ) . (2.9)To cope with the drop in index (dimension) in (2.4), we need the Feldheim-Vilenkin integral [ , (2.11)], [ , p.315], [ ], W λn ( x ) = (cid:20) λ + )Γ( ν + )Γ( λ − ν ) (cid:21) (cid:90) u ν (1 − u ) λ − ν − · [ x − x u + u ] n W νn (cid:18) x √ x − x u + u (cid:19) du. (2.10)3. The result
We can now formulate our result.
Theorem 3.1.
For r ∈ ( − , , the sum of the Askey-Fitch series (2.4) above is given bythe integral (3.1) below: (cid:90) H λν ( du ) (cid:90) − G ν − ( dv ) (cid:2) − r ( x − x u + u ) (cid:3) I ν +1 , (3.1) where I is given by I := 1 − r · xy + uv √ − x (cid:112) − y √ x − x u + u + ( xy + uv √ − x (cid:112) − y ) ( x − x u + u ) . (3.2) Moreover, this holds also for r = 1 unless ν < λ ≤ ν + 1 .Proof. We sum the series by reducing it to the generating function (2.1). There are twosteps: reduction of λ to ν by the Feldheim-Vilenkin integral (2.10) and reduction of two W n terms to one by Gegenbauer’s multiplication theorem (2.9).We follow [ ]. As there, we may substitute for W λn ( x ) from (2.10) into the series (2.4) andintegrate termwise, rewriting (2.4) as (cid:90) H λν ( du ) ∞ (cid:88) n =0 ω νn ( r [ x − x u + u ] ) n · W νn ( y ) W νn (cid:18) x √ x − x u + u (cid:19) . (3.3)We use Gegenbauer’s multiplication theorem (2.9) with r (cid:55)→ r (cid:112) x − x u + u , and replace the product of W νn factors in the above, at the cost of another integration over G ν − ( dv ), by a single W νn term, with argument xy (cid:112) x − x u + y + v (cid:112) − y . (cid:114) − x x − x u + u = xy + uv √ − x (cid:112) − y √ x − x u + u . (3.4)The integrand is now of the form (cid:80) ω νn r n W νn ( · ), and the result now follows from the generatingfunction (2.1).age 4 of 7 N. H. BINGHAM AND TASMIN L. SYMONS
Figure 1.
Numerical evaluation of series (3.1) for λ = 3 . . ν = 0 . . This result completes and complements the work in [ ] and [ ] by displaying the dependenceon the higher index λ in a structurally revealing way: for simplicity, let r = 1 so that I = (cid:32) − xy + uv √ − x (cid:112) − y √ x − x u + u (cid:33) , (3.5)and (3.1) is given by (cid:90) H λν ( du ) (cid:90) − G ν − ( dv ) 1 − ( x − x u + u ) I ν +1 . (3.6)Using the definition of H λν and the probability measure G ν + and simplifying, (3.1) becomes2 √ π (cid:90) Γ( λ + )Γ( λ − ν ) u ν (1 − u ) λ − ν − (cid:90) − (1 − v ) ν − (cid:20) − ( x − x u + u ) I ν +1) (cid:21) dvdu. (3.7)Note that the higher index λ occurs only in the outer integral. Moreover, the interactionsbetween the indexes in the outer integral occurs only in the Gamma function Γ( λ − ν ) and thepower λ − ν − / − u ). 4. Dimension walks
Write P ν for the class of functions f on [ − ,
1] which are mixtures of W νn , i.e., of the form f ( x ) = ∞ (cid:88) n =0 a n W νn ( x ) , (cid:88) a n = 1 , a n ≥ a = { a n } ∞ (the ultraspherical series converges uniformly as | W νn ( x ) | ≤ P ν are decreasing in ν ∈ [0 , ∞ ], and are continuous in ν , in that (cid:92) {P µ : 0 ≤ µ < ν } = P ν , (cid:91) {P µ : ν < µ ≤ ∞} = P ν EPRESENTATIONS OF ULTRASPHERICAL POLYNOMIALS
Page 5 of 7([ , Th. 1]). While the parameters λ , ν > S d the d -sphere – the unit sphere in Euclidean ( d + 1)-space R d +1 ,a d -dimensional Riemannian manifold – the relevant index for the ultraspherical polynomial is ν , where ν = 12 ( d − . With ν < λ as above, the higher dimension corresponding to λ will be written d (cid:48) (so λ = ( d (cid:48) − ], [ ], the passage from λ to ν < λ corresponds to projection from the d (cid:48) -sphere to the d -sphere. The limiting case ν = ∞ gives W ∞ n ( x ) = x n , and P ∞ is the class ofprobability generating functions, or the class of positive definite functions on the unit spherein Hilbert space ([ , Lemma 2], [ ]).Covariance functions on spheres are very valuable in applications to Planet Earth (see § ], [ ]. The one-step walks in [ ] are based on the Riemann-Liouvilleoperators, but lack the highly desirable semi-group property , in which passage from λ to ν andthen ν to µ is the same as passage from λ to µ directly. Theorem 4.1.
For f ∈ P ν as in (4.1), f ( x ) = ∞ (cid:88) n =0 a n W λn ( x ) ∈ P λ . (4.2) Proof. (cid:90) − f ( y ) M λν ( x ; dy ) = (cid:90) − ∞ (cid:88) n =0 a n W νn ( y ) M λν ( x ; dy ) (4.3)= ∞ (cid:88) n =0 a n (cid:90) − W νn ( y ) M λµ ( x ; dy ) (4.4)= ∞ (cid:88) n =0 a n W λn ( x ) ∈ P ( S d (cid:48) ) , (4.5)interchange of the sum and integral in 4.4 being justified by the uniform convergence of theSchoenberg expansion. Corollary 4.2.
The operation of passing from f ( x ) ∈ P ν to (cid:82) − f ( y ) M λν ( x, dy ) ∈ P λ inthe theorem has the semigroup property.Proof. The mixture coefficients a n are unchanged by this operation, and so remainunchanged under further operations of the same type.5. Complements
Hypergroups and symmetric spaces.
Hypergroups are ‘locally compact spaces with a group-like structure on which the boundedmeasures convolve in a similar way to that on a locally compact group’, to quote from theage 6 of 7
N. H. BINGHAM AND TASMIN L. SYMONS standard work on this important subject, [ , p.1]. The probabilistic setting of random walkson spheres [ , p.196-197] that inspired both [ ] and this paper, its sequel, is in hypergrouplanguage that of the Bingham (or Bingham-Gegenbauer) hypergroup. This in turn was inspiredby Kingman’s work on random walks with spherical symmetry [ ], which gives the Kingman(or Kingman-Bessel) hypergroup. The theory for spheres and for spherical symmetry give theprototypical examples of symmetric spaces of rank one of compact type (constant positivecurvature) and of Euclidean type (zero curvature); these are complemented by the case ofconstant negative curvature, the hyperbolic or Zeuner hypergroups [ ]. For background onsymmetric spaces we refer to Helgason [ ], for spaces of constant curvature to Wolf [ ], andfor compact symmetric spaces to Askey and Bingham [ ].We note that the Kingman situation (Euclidean space with spherical symmetry) may berecovered from the spherical one here by letting the radius of the sphere tend to infinity. TheBessel functions in the Kingman theory arise from radialisation of the Fourier transform inEuclidean space under spherical symmetry [ , II.7].5.2. Gaussian processes, path properties, Tauberian theorems.
The positive definite functions in the classes P ν of § a = { a n } (the angular power spectrum ) of the Schoenberg expansion coefficients above. In particular,the rate of decay of the a n governs the path properties: the faster the decay, the smootherthe paths. For details, see [ ]. Crucial here is Malyarenko’s theorem [ , Ch. 4]. This restson a Tauberian theorem of the first author [ ], which in turn derives from work of Askeyand Wainger [ ]. Here it is necessary to move from the one-parameter family of ultrasphericalpolynomials W νn to the two-parameter family of Jacobi polynomials J α,βn containing it ([ , Ch.6], [ , Ch. IV]).5.3. Sphere cross line.
The motivation for much of the interest in positive definite functions on spheres derives fromits applications in geostatistics. Here one has both spatial dependence and temporal evolution,and so one is dealing with geotemporal processes. For background here, see e.g. [ ], [ ]. Postscript
To close, the first author takes pleasure in noting the half-century between Part I [ ] (whichderives from his own PhD of 1969) and the present Part II (which derives from the secondauthor’s PhD of 2020). We both take pleasure in dedicating the paper to the memory of DickAskey, whose influence pervades it. Dick was a famous expert on special functions, but wasinterested in their applications, including those to probability. When [ ] was written, he used todine out by saying, with tongue in cheek, “I’ve just written a paper with Bingham on Gaussianprocesses – whatever they are.” References G. E. Andrews, R. Askey, and R. Roy.
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N. H. BinghamDepartment of MathematicsImperial College LondonSouth Kensington CampusLondon, SW7 1AZUK [email protected]