Advances in Pure Mathematics, 2019, 9, 15-42 http://www.scirp.org/journal/apm ISSN Online: 2160-0384 ISSN Print: 2160-0368 Associated Hermite Polynomials Related to Parabolic Cylinder Functions Alfred Wünsche Max-Planck-Group “Nonclassical Radiation” at Institut für Physik, Humboldt-Universität Berlin, Berlin, Germany How to cite this paper: Wünsche, A. Abstract (2019) Associated Hermite Polynomials ν Related to Parabolic Cylinder Functions. In analogy to the role of Lommel polynomials R n ( z) in relation to Bessel Advances in Pure Mathematics, 9, 15-42. https://doi.org/10.4236/apm.2019.91002 functions Jν ( z) the theory of Associated Hermite polynomials in the scaled ν form Hen ( z) with parmeter ν to Parabolic Cylinder functions Dν ( z) is Received: November 26, 2018 Accepted: January 20, 2019 developed. The group-theoretical background with the 3-parameter group of Published: January 23, 2019 motions M (2) in the plane for Bessel functions and of the Heisen- Copyright © 2019 by author(s) and berg-Weyl group W (2) for Parabolic Cylinder functions is discussed and Scientific Research Publishing Inc. compared with formulae, in particular, for the lowering and raising operators This work is licensed under the Creative and the eigenvalue equations. Recurrence relations for the Associated Her- Commons Attribution International License (CC BY 4.0). mite polynomials and for their derivative and the differential equation for http://creativecommons.org/licenses/by/4.0/ them are derived in detail. Explicit expressions for the Associated Hermite Open Access polynomials with involved Jacobi polynomials at argument zero are given and by means of them the Parabolic Cylinder functions are represented by two such basic functions. Keywords Bessel Functions, Lommel Polynomials, Parabolic Cylinder Functions, Associated Hermite Polynomials, Jacobi polynomials, Recurrence Relations, Lowering and Raising Operators, Heisenberg-Weyl Group, Motion Group of Plane, Irreducible Representations 1. Introduction Let us motivate our intentions by an analogy of Bessel functions Jν ( z) to Pa- rabolic Cylinder functions Dν ( z) . Both sets of functions satisfy a certain second-order differential equation and a certain 3-term recurrence relation. The 3-term recurrence relation for the Bessel functions in the form Jν +n ( z) pro- DOI: 10.4236/apm.2019.91002 Jan. 23, 2019 15 Advances in Pure Mathematics A. Wünsche vides the possibility to express it successively by the sum of two Bessel functions with lower indices Jν +m ( z) and Jν +−m 1 ( z) , ( mn< ), and we may continue this up to the case where we have presented Jν +n ( z) by a superposition of two Bes- sel functions Jν ( z) and Jν −1 ( z) considered as our “basic” functions of the form νν+1 Jν+nn( z) =−=∈ R( zz) J νν( ) R n−−11( z) J( zn) ,( 0,1,2, ,ν ) , (1.1) ν −1 with coefficients R n ( z) which are polynomials of n-th degree of variable z called Lommel polynomials. Their explicit form is known [1] (cited according to Watson [2]). We find this in Watson [2] (from p. 294 on in very detailed form) and in Bateman and Erdélyi [3] (chap. 7.5.2, p. 43) with the explicit formula for the Lommel polynomials1 ( [µ] means integer part of µ ) n k 2 nk−2 ν (−1) (nk −) !( n +−−ν k 1!) 2 R.n ( z) ≡ ∑ (1.2) k =0 kn!( − 2 k) !(ν +− k 1!) z Whereas Lommel derives his polynomials by a somewhat cumbersome induc- tion ([2], chap. 9.61.) Watson derives the Lommel polynomials from some (Laurent) series of products of Bessel functions that according to his statement is simpler. Furthermore, he discusses in chap. 9.7 (from p. 303 on) a related n zz2 function gznn,,νν≡ R ( ) which in similar form was introduced by 42 Hurwitz [4] (cited on page 302 in [2]). By an analogous process one may relate the Parabolic Cylinder functions Dν +n ( z) to the sum of two Parabolic Cylinder functions Dν ( z) and Dν −1 ( z) considered as our “basic” functions of the form νν+1 Dνν+nn( z) = He( zz) D( ) −ν Hen−−11( z) D ν( z) , (1.3) ν with polynomials Hen ( z) of n-th degree in variable z which possess the expli- cit form n k 2 (nk−)!( − 1) k (ν −+ 1!j) ν nk−2 Hen ( zz) = ∑∑− −−ν kj kj=0 (nk2) !( 1!) =0 jk!( − j) !2 (1.4) n 2 (nk− )! (−−ννkn, +− k) nk−2 = ∑ Pk ( 0,) z k =0 (nk− 2!) (αβ, ) where Pk (u) denotes the Jacobi polynomials introduced in this form by Szegö [5] and here taken for argument u = 0 . The polynomials (1.4) are asso- ciated to the Hermite polynomials in a scaled form and are for ν = 0 identical with the scaled Hermite polynomials usually denoted by Hen ( z) that means 0 Henn( zz) ≡ He ( ) . We call them Associated Hermite polynomials (scaled for applications in connection with the Parabolic Cylinder functions Dν ( z) ). They 1We change slightly the standard notation of the Lommel polynomials [1] [2] [3] according to ν RRnn,ν ( zz) → ( ) for reason which is better to feel after the development of the analogous formal- ism for Parabolic Cylinder functions. DOI: 10.4236/apm.2019.91002 16 Advances in Pure Mathematics A. Wünsche were introduced together with some of their basic relations in [6]. One of the possible applications of Parabolic Cylinder functions is in quantum mechanics for the calculation of eigenfunctions of squeezing and rotation oper- ators in the two-dimensional phase plane that means for the general quadratic combinations of a pair of boson annihilation and creation operators and, more generally, for several pairs of boson operators. This is also connected with the reduction to two possible normal forms and one degenerate form of quadratic combinations of such operators. For squeezing operators the eigenfunctions are not normalizable and up to now are little known. The notions “Associated Hermite polynomials” and “Generalized Hermite polynomials” are not used in fully unique sense in literature, e.g., [7] [8]. How- ν ever, our notation Hen ( z) for them seems to be unique and specifies them. Our “Associated Hermite polynomials” are related to the Parabolic Cylinder functions in the analogous way as Lommel polynomials are related to Bessel functions. They are, in general, not orthogonal polynomials and satisfy a 4-th order differential equation. In the following Sections we develop more systematically some formalism for the Parabolic Cylinder functions in connection with their Associated Hermite polynomials. Later, after this we will come back again to the analogies between Bessel functions and Parabolic Cylinder functions with discussion of the group-theoretical background. 2. The Parabolic Cylinder Functions in the Form Dν (z) The Weber equation with parameter ν ∈ ∂22z 1 − ++νν = 2 yz( ;) 0, (2.1) ∂z 42 with important application in physics (e.g., quantum mechanics) is satisfied, for example, by the following two independent solutions zz22 ν 1 yz1(ν ;) =−− exp 11 F ; ; , 4 22 2 zz2213−ν yz2(ν ;) = exp − z11 F ; ; , (2.2) 4 2 22 with the Wronski determinant ∂∂ Wyyzyzyzyzyz( ,)(νν ;) ≡−=( ;) ( νν ;) ( ;) ( ν ;) 1. (2.3) 12 1 ∂∂zz2 2 1 It is independent of variable z due to differential Equation (2.1) with no first-order derivative and with a second-order derivative without a coefficient in front depending on z as it is easily to derive. In addition, it is independent on parameter ν that is a special property of the two solutions (2.2). Due to inde- pendence of z we obtain the Wronski determinant (2.3) setting, for example, z = 0 . DOI: 10.4236/apm.2019.91002 17 Advances in Pure Mathematics A. Wünsche From the two linearly independent solutions yz1 (ν ; ) and yz2 (ν ; ) one constructs by superposition as definition yz(ν ;D) ≡ ν ( z) the following solu- tion of the Weber Equation (2.1) 1 − ! ν 22 ν 2 zz 2 1 Dν ( z) ≡− 2 exp 11F;; − 4 1+ν 22 2 − ! 2 3 − ! 2 zz2 13−ν + 11F ;; 2 ν 2 22 −−1! 2 (2.4) 1 − ! ν 22 +ν 2 zz 2 11 = 2 exp 11F;; − 4 1+ν 22 2 − ! 2 3 − ! 2 z 2 ν 3 z + 11F1+− ; ; , 2 ν 22 2 −1!− 2 where we used in the second representation the Kummer transformation of the Confluent Hypergeometric function (e.g., [9], (chap. (6.3, Equation (7))) z 11F(acz ;;) = e11 F( c −− ac ;; z) , (2.5) 13 and with −=! π, −=− !2π . A second independent solution of the 22 Weber equation can be constructed in various ways from superpositions of the kind fz(να;) = Dνν( z) + β D( −+ z) γ D−− ν11( i z) + δ D−− ν( − i, z) (2.6) with coefficients (αβγδ, ,,) but maximally only two of the four basic functions (Dνν(±±zz) ,D−−1 ( i )) can be linearly independent solutions of the Weber equa- tion. For example ν ! π π Dν( z) =expiνν D−− νν11( i zz) +− expi D−−( − i) , (2.7) 2π 22 is such a relation between three of these functions [3] (chap. 8.2. (6)). Instead of Dν ( z) other authors use a fully equivalent form of basic Parabol- ic Cylinder function with notation U,(µ z) related to Dν ( z) as follows (e.g., Temme [10] in chap. 12 of NIST Handbook [11] and also J. Miller [12] in the older Handbook by the editors Abramowitz and Stegun [13]) 1 Dν ( z) ≡ U −−νµ ,, z ⇔ U,( zz) ≡ D1 ( ) . (2.8) −−µ 2 2 A few but very important special cases of these definitions which can be also DOI: 10.4236/apm.2019.91002 18 Advances in Pure Mathematics A. Wünsche expressed by other Special functions are [10] (p. 309) 3 zzz2 22 =−= + D1( zz) U( 1, ) KK,31 2 22π 4444 zz2 D11( zz) = U( 0,) = K , − 242π 4 3 zz2 22 z D3( zz) = U( 1,) = K31− K , (2.9) − 2 2π 4444 and (e.g., [10], p.