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The Hypergeometric Functions Approach to the Connection Problem for the Classical Orthogonal Polynomials

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The Hypergeometric Functions Approach to the Connection Problem for the Classical Orthogonal Polynomials Technical Report Inst. of Computer Sci., Univ. of Wroc law Feb. 1998; rev. Aug. 2003 DOI: 10.13140/RG.2.1.2993.3209 http://www.ii.uni.wroc.pl/~sle/publ.html THE HYPERGEOMETRIC FUNCTIONS APPROACH TO THE CONNECTION PROBLEM FOR THE CLASSICAL ORTHOGONAL POLYNOMIALS STANISLAW LEWANOWICZ Abstract. Let fPkg and Qk be any two sequences of classical orthogonal polynomials. Using theorems of the theory of generalized hypergeometric functions, we give closed-form expressions as well as recurrence relations for the coefficients an;k in the connection equation Pn Qn = k=0 an;kPk (n 2 N). 1. Introduction Let fPk(x)g and fQk(x)g be two systems of the classical orthogonal polynomials, i. e. associated with the names of Jacobi, Laguerre, Hermite and Bessel. We are looking for the coefficients an;k in n X (1.1) Qn = an;kPk; k=0 called connection coefficients. An analogous problem is formulated for fPk(x)g and fQk(x)g being systems of the classical orthogonal polynomials of a discrete variable, i. e. associated with the names of Charlier, Meixner, Krawtchouk and Hahn. Two type of results are met in a vast literature of this subject: closed-type formulae, or recurrence relations (usually in k) for an;k (see [2]{[5], [10], [11], [13], [14], [16]{[25], [29]{ [33], [36]). In the present paper, we show that both types of information on an;k may be obtained using theorems of the theory of generalized hypergeometric functions. 2. Some results on generalized hypergeometric functions Throughout this section, the letters p, q, r, s, t, u and n stand for non-negative integers. We shall use the Pochhhammer's symbol (α)k = Γ(k + α)=Γ(α): The definition for the generalized hypergeometric function is 1 k [ap] X (ap)kx (2.1) pFq x = [bq] (bq)kk! k=0 where the symbols [ap] and [bq] denote sets a1; a2; : : : ; ap and b1; b2; : : : ; bq of complex param- eters, respectively, such that −bj 62 N0 and ai 6= bj for 1 ≤ i ≤ p and 1 ≤ j ≤ q, and where 2 STANISLAW LEWANOWICZ the above contracted notation will be used troughout the paper p q Y Y f(ap) = f(ai); f(bq) = f(bj); i=1 j=1 f being a given function. Lemma 2.1. (2.2) n k −n; [ap]; [cr] X n (ap)k(αt)kz p+r+1Fq+s z! = [bq]; [ds] k (bq)k(βu)k(k + λ)k k=0 k − n; [k + ap]; [k + αt] −k; k + λ, [cr]; [βu] × p+t+1Fq+u+1 z r+u+2Fs+t ! : 2k + λ + 1; [k + bq]; [k + βu] [ds]; [αt] Proof. Eq. (2.2) is a specialized form of a much more general result in [26, Vol. II, x9.1, Eq. (13)]. Lemma 2.2. n k −n; [ap]; [cr] X n (ap)k(αt)kz (2.3) p+r+1Fq+s z! = [bq]; [ds] k (bq)k(βu)k k=0 k − n; [k + ap]; [k + αt] −k; [cr]; [βu] × p+t+1Fq+u z r+u+1Fs+t ! : [k + bq]; [k + βu] [ds]; [αt] Proof. Eq. (2.3) is a specialized form of a much more general result in [26, Vol. II, x9.1, Eq. (27)]. Lemma 2.3 ( Wimp [34]; see also Wimp [35, App. C]; or Luke [26, Vol. II, x12.4]). Let β, µ, [φP ], [ Q] be such that none of the quantities β + 1, φi, j and µ are negative integers or zero. Let w 2 C n f0g and let φi = β + 1 for i = P + 1. Then the functions (φP )k k [k + φP +1] (2.4) Uk(w) = w P +1FQ+1 w (k ≥ 0); (k + µ)k( Q)k 2k + µ + 1; [k + Q] satisfy the difference equation θ+1 X (2.5) fAm(k; θ + 1)w + Bm(k; θ + 1)g Uk+m(w) = 0; m=0 where θ := max(P; Q + 1), (2.6) A0(k; R) ≡1; m (2k + µ)m(k + β + 1)m (2.7) Am(k; R) =(−1) m!(k + µ)m −m; 2k + µ + m; [k + φP +1 + 1] × P +3FP +2 1 (1 ≤ m ≤ R); 2k + µ + R + 1; [k + φP +1] CONNECTION PROBLEM FOR THE CLASSICAL ORTHOGONAL POLYNOMIALS 3 (2.8) B0(k; R) =BR(k; R) ≡ 0; m (2k + µ)m+1(k + β + 1)m[k + Q] (2.9) Bm(k; R) =(−1) (m − 1)!(k + µ)m[k + φP +1] 1 − m; 2k + µ + m + 1; [k + Q + 1] × Q+2FQ+1 1 (1 ≤ m ≤ R): 2k + µ + R + 1; [k + Q] Functions Uk(w) do not satisfy any other difference equation of type (2.5), with Am and Bm independent of w, of order ≤ θ + 1. In the special case of P = Q + 1 and w = 1, the above result can be refined in the following sense. Lemma 2.4 (Lewanowicz [15]). Functions (φQ+1)k [k + φQ+2] (2.10) Uk(1) = Q+2FQ+1 1 (k + µ)k( Q)k 2k + µ + 1; [k + Q] satisfy the difference equation θ X (2.11) fAm(k; θ) + Bm(k; θ)g Uk+m(1) = 0; m=0 of order θ := Q + 1, notation being that of (2.7)-(2.9). We have the following confluent version of the result given in Lemma 2.3. Lemma 2.5. Let β, [φP ], [ Q] be such that none of the quantities β + 1, φi, j and µ are negative integers or zero. Let w 2 Cnf0g and let φi = β +1 for i = P +1. Then the functions (φP )k k [k + φP +1] (2.12) Vk(w) = w P +1FQ w (k ≥ 0); ( Q)k [k + Q] satisfy the difference equation # X (2.13) fCm(k)w + Dm(k)g Vk+m(w) = 0; m=0 where # := max(P; Q) + 1, C0(k) ≡ 1, D0(k) ≡ 0, and m (k + β + 1)m −m; [k + φP +1 + 1] (2.14) Cm(k) =(−1) P +2FP +1 1 ; m! [k + φP +1] m (k + β + 1)m(k + Q) 1 − m; [k + Q + 1] (2.15) Dm(k) =(−1) Q+1FQ 1 (m − 1)!(k + φP +1) [k + Q] for m = 1; 2;:::;#. Proof. Replacing in (2.5) w by µw and passing to the limit with µ ! 1, and making use of the confluence principle [k + φP +1] [k + φP +1] lim P +1FQ+1 µw = P +1FQ w µ!1 [k + Q]; 2k + µ + 1 [k + Q] (see [26, Vol. I, x3.5]), we obtain the equation θ+1 X fCm(k)w + Dm(k)g Vk+m(w) = 0; m=0 4 STANISLAW LEWANOWICZ with θ = max(P; Q + 1). If P > Q then θ = P , and by [26, Vol.I, Eqs. 2.9(14), (15)] Cθ+1(k) 6≡ 0, Dθ+1(k) ≡ 0. If P ≤ Q then θ = Q + 1, and by the above-cited equations Cθ+1(k) = Dθ+1(k) ≡ 0, Dθ(k) 6≡ 0. Hence the result. In the special case of P = Q and w = 1, the above result can be refined in the following sense. Lemma 2.6. The functions (φQ+1)k [k + φQ+1] (2.16) Vk(1) = Q+1FQ 1 (k ≥ 0) ( Q)k [k + Q] satisfy the difference equation Q X (2.17) fCm(k) + Dm(k)g Vk+m(w) = 0; m=0 notation being that of (2.14)-(2.15) with P = Q. Proof. In Lemma 2.5, let P := Q and w := 1. It suffices to show that the last term of the sum in (2.13) vanishes, so that the order of the difference equation reduces to Q. Indeed, using [26, Vol. I, Eq. 2.9(14)], we obtain (k + β + 1)Q+1 CQ+1 = = −DQ+1: [k + φQ+1] 3. Classical polynomials orthogonal on an interval Let fPk(x)g and fQk(x)g be two systems of the classical orthogonal polynomials, i. e. associated with the names of Jacobi, Laguerre, Hermite and Bessel. Let us look for the coefficients an;k in n X (3.1) Qn(c x) = an;k(c)Pk(x)(c 6= 0); k=0 slightly generalizing equation (1.1). The following hypergeometric series representation of these polynomials is well known: (α,β) n n + β −n; n + α + β + 1 1 + x (3.2) P (x) =(−1) 2F1 ; n n β + 1 2 α n −n (3.3) L (x) =(α + 1)n(−1) 1F1 x ; n α + 1 n −n=2; −n=2 + 1=2 1 (3.4) Hn(x) =(2x) 2F0 − ; | x2 α −n; n + α + 1 x (3.5) Y (x) = 2F0 − : n | 2 Shifted Jacobi polynomials are given by (α,β) (α,β) (3.6) Rn (x) = Pn (2x − 1): CONNECTION PROBLEM FOR THE CLASSICAL ORTHOGONAL POLYNOMIALS 5 3.1. Explicit forms for the connection coefficients. The above results allow to obtain connection formulae between classical orthogonal polynomials. For instance, the following expansions hold: 8 n > β X BB α >Yn (c x) = an;k (c)Yk (x); <> (3.7) k=0 ! n(n + β + 1) k − n; k + n + β + 1 > BB k k > an;k (c) = c 2F1 c ; :> k (k + α + 1)k 2k + α + 2 8 n > γ X LL α >Ln(c x) = an;k(c)Lk (x); <> (3.8) k=0 k ! > LL n n (−c) k − n; k + α + 1 > an;k(c) = (−1) (γ + 1)n 2F1 c ; :> k (γ + 1)k k + γ + 1 8 n > γ X LB α >Ln(c x) = an;k (c)Yk (x); <> (3.9) k=0 k ! > LB n n (α + 1)k(−2c) k − n > an;k (c) = (−1) (γ + 1)n 1F1 −2c ; :> k (γ + 1)k k + γ + 1 8 n > γ X BJ (α,β) >Yn (cx) = an;k(c)R (x); > k > k=0 > k < BJ (−n)k(n + γ + 1)k c (3.10) an;k(c) = − > (k + α + β + 1)k 2 > ! > k + β + 1; k − n; k + n + γ + 1 c > × 3F1 − : :> 2k + α + β + 2 2 8 n (γ,δ) X >R (c x) = aJB(c)Y α(x); > n n;k k > > k=0 < JB n n (k + δ + 1)n−k(n + γ + δ + 1)k k (3.11) an;k(c) = (−1) (−2c) > k n!(k + α + 1)k > ! > k − n; k + n + γ + δ + 1 > × 2F2 −2c ; :> 2k + α + 2; k + δ + 1 8 n (γ,δ) X (α,β) >P (x) = aJJ P (x); > n n;k k > > k=0 < JJ n n + δ (−n)k(n + γ + δ + 1)k (3.12) an;k = (−1) > n (δ + 1)k(k + α + β + 1)k > ! > k − n; k + β + 1; k + n + γ + δ + 1 > × 3F2 1 ; :> 2k + α + β + 2; k + δ + 1 6 STANISLAW LEWANOWICZ For instance, let us prove (3.12).
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