Orthogonal sequences constructed from quasi-orthogonal ultraspherical polynomials

Abstract

Let $\displaystyle \{x_{k,n-1}\}_{k=1}^{n-1}$ and $\displaystyle \{x_{k,n}\}_{k=1}^{n},$$n \in \mathbb {N}$, be two sets of real, distinct points satisfying the interlacing property $x_{i,n}<x_{i,n-1}< x_{i+1,n}, i = 1,2,\dots ,n-1$. In [15], Wendroff proved that if $p_{n-1}(x) = \displaystyle \prod \limits _{k=1}^{n-1} (x-x_{k,n-1})$ and $p_{n}(x) = \displaystyle \prod \limits _{k=1}^{n} (x-x_{k,n})$, then p_n− 1 and p_n can be embedded in a non-unique monic orthogonal sequence $\{p_{n}\}_{n=0}^{\infty }.$ We investigate a question raised by Mourad Ismail as to the nature and properties of orthogonal sequences generated by applying Wendroff’s Theorem to the interlacing zeros of $C_{n-1}^{\lambda }(x)$ and $(x^{2}-1) C_{n-2}^{\lambda }(x)$, where $\{C_{k}^{\lambda }(x)\}_{k=0}^{\infty }$ is a sequence of monic ultraspherical polynomials and − 3/2 < λ < − 1/2, λ≠ − 1. We construct an algorithm for generating infinite monic orthogonal sequences $\{D_{k}^{\lambda }(x)\}_{k=0}^{\infty }$ from the two polynomials $D_{n}^{\lambda } (x): = (x^{2}-1) C_{n-2}^{\lambda } (x)$ and $D_{n-1}^{\lambda } (x): = C_{n-1}^{\lambda } (x)$, which is applicable for each pair of fixed parameters n, λ in the ranges $n \in \mathbb {N}, n \geq 5$ and λ > − 3/2, λ≠ − 1,0,(2k − 1)/2, k = 0,1,…. We plot and compare the zeros of $D_{m}^{\lambda } (x)$ and $C_{m}^{\lambda } (x)$ for selected choices of $m \in \mathbb {N}$ and a range of values of the parameters λ and n. For − 3/2 < λ < − 1, the curves that the zeros of $D_{m}^{\lambda } (x)$ and $C_{m}^{\lambda } (x)$ approach are substantially different for large values of m. In contrast, when − 1 < λ < − 1/2, the two curves have a similar shape while the curves are almost identical for λ > − 1/2.

Appendices

Appendix 1. Mathematica script for construction of an orthogonal sequence $\{D_j^{\lambda }\}_{j=0}^{N+K-2}$, N ≥ 5, K ≥ 3

(* Mathematica script to construct the orthogonal sequence {D_j^\ \[Lambda]}, where 0\[LessEqual]j\[LessEqual]NN+KK-2, from the \ starting polynomials D_NN^\[Lambda](x)=(x^2-1) C_(NN-2)^\[Lambda](x) \ and D_(NN-1)^\[Lambda]=C_(NN-1)^\[Lambda](x) *) ClearAll; NN = 5; KK = 11; \[Lambda] = -5/4; \[Sigma] = 2; a = (4 (2 + \[Lambda]))/(3 (3 + 2 \[Lambda])); (*Compute DNm1=D_N^\[Lambda]*) KN = Coefficient[(x^2 - 1) JacobiP[ NN - 2, \[Lambda] - 1/2, \[Lambda] - 1/2, x], x^NN]; DN = Simplify[ 1/KN (x^2 - 1) JacobiP[NN - 2, \[Lambda] - 1/2, \[Lambda] - 1/2, x]]; (*Compute DNm1=D_(N-1)^\[Lambda]*) KNm1 = Coefficient[ JacobiP[NN - 1, \[Lambda] - 1/2, \[Lambda] - 1/2, x], x^(NN - 1)]; DNm1 = Simplify[ 1/KNm1 JacobiP[NN - 1, \[Lambda] - 1/2, \[Lambda] - 1/2, x]]; (* Construct D_j^\[Lambda] for j=N-2,N-3,...,0, in the array \ DjLowerDegrees *) \[Beta]2 = Table[0, {j, 1, NN}]; (*\[Beta]2[[1]] must be zero*) ell = Table[0, {j, 1, NN}]; (*ell[[1]] will not contain any information*) \[Beta]2[[NN]] = Simplify[Coefficient[DN, x^(NN - 2)]]; \[Beta]2[[NN - 1]] = Simplify[Coefficient[DNm1, x^(NN - 3)]]; ell[[NN]] = Simplify[\[Beta]2[[NN - 1]] - \[Beta]2[[NN]]]; Clear[DjLowerDegrees]; DjLowerDegrees = Table[0, {k, 0, NN}]; DjLowerDegrees[[(NN) + 1]] = DN; DjLowerDegrees[[(NN - 1) + 1]] = DNm1; Do[{j = k + 2; DjLowerDegrees[[(NN - j) + 1]] = Simplify[-1/ ell[[NN - j + 2]] (DjLowerDegrees[[(NN - j + 2) + 1]] - x DjLowerDegrees[[(NN - j + 1) + 1]])]; If[j == NN - 2, \[Beta]2[[2]] = Simplify[DjLowerDegrees[[(2) + 1]] - x^2], \[Beta]2[[NN - j]] = Simplify[ Coefficient[DjLowerDegrees[[(NN - j) + 1]], x^(NN - j - 2)]]]; ell[[NN - j + 1]] = \[Beta]2[[NN - j]] - \[Beta]2[[NN - j + 1]]; }, {k, 0, NN - 2}]; (* Construct D_j^\[Lambda] for j=N+1,N+2,...,N+M, in the array \ DjUpperDegrees *) Clear[DjUpperDegrees]; DjUpperDegrees = Table[0, {k, 1, KK}];(* DjUpperDegrees [[k]]=D_{N+k-2}^\[Lambda]*) DjUpperDegrees[[1]] = DNm1; DjUpperDegrees[[2]] = DN; ellNp1 = Simplify[(a DN /. x -> a)/(\[Sigma] DNm1 /. x -> a)]; DjUpperDegrees[[3]] = Simplify[x DN - ellNp1 DNm1]; ellNpj = (\[Sigma] - 1)/\[Sigma]^2 a^2; Do[{ DjUpperDegrees[[j + 2]] = Simplify[ x DjUpperDegrees[[j + 1]] - ellNpj DjUpperDegrees[[j]]]; }, {j, 2, KK - 2}]; (*Print out DjLowerDegrees in standard form*) Print["N = ", NN, ", \[Lambda]=", \[Lambda], ", KK=", KK, ", a=", a] Do[{CoefRulesDj = Simplify[CoefficientRules[ DjLowerDegrees[[j]], {x}]]; DNPrint = TraditionalForm[FromCoefficientRules[CoefRulesDj, {x}], ParameterVariables :> {\[Lambda]}]; Print["D_", j - 1, "^\[Lambda] = ", TraditionalForm[DNPrint, ParameterVariables :> {\[Lambda]}], "."]}, {j, 1, NN + 1}]; (*Print out DjUpperDegrees in standard form*) Do[{CoefRulesDj = Simplify[CoefficientRules [DjUpperDegrees[[j]], {x}]]; DNPrint = TraditionalForm[FromCoefficientRules[CoefRulesDj, {x}], ParameterVariables :> {\[Lambda]}]; Print["D_", NN + j - 2, "^\[Lambda] = ", TraditionalForm[DNPrint, ParameterVariables :> {\[Lambda]}], "."];}, {j, 1, KK}];

Appendix: 2. The first 11 terms of the polynomial sequence $\{D_m^{\lambda }\}_{m=0}^{\infty }$ with n = 5, k = 5, σ = 2, and $a=\frac {4(2+\lambda )}{3(3+2\lambda )}$

In this section we compute the first several terms of the sequence $\{D_m^{\lambda }\}_{m=0}^{\infty }$ for a specific choice of parameters. The purpose of this example is to get a glimpse of the symbolic expressions for the polynomials for a general (not fixed) value of λ. The example also provides a way to verify correctness of programming for those readers who are interested to implement the algorithm developed in Section 4, using programming environment of their choice. We thus provide the first 11 terms of the polynomial sequence $\{D_m^{\lambda }\}_{m=0}^{\infty }$ obtained using the algorithm developed in Section 4 with n = 5, k = 5, σ = 2, and $a=\frac {4(2+\lambda )}{3(3+2\lambda )}$, where $\lambda \in (-\frac {3}{2}, +\infty )$, λ≠− 1, 0 and λ≠(2k − 1)/2, k = 0, 1, 2,…:

$$ \begin{array}{@{}rcl@{}} &&D_{0}^{\lambda}(x) = 1,\\ &&D_{1}^{\lambda}(x) = x,\\ &&D_{2}^{\lambda}(x) = x^{2}-\frac{2 \lambda^{2}+7 \lambda+9}{2 (2 \lambda^{3}+7 \lambda^{2}+9 \lambda+6)},\\ &&D_{3}^{\lambda}(x) = x^{3}-\frac{3 (2 \lambda+5)}{2 (2 \lambda^{2}+7 \lambda+9)}x,\\ &&D_{4}^{\lambda}(x) = x^{4}-\frac{3 }{\lambda+3}x^{2}+\frac{3}{4 (\lambda^{2}+5 \lambda+6)},\\ &&D_{5}^{\lambda}(x) = x^{5}-\frac{(2 \lambda+7) }{2 \lambda+4}x^{3}+\frac{3 }{2 \lambda+4} x, \end{array} $$

$$ \begin{array}{@{}rcl@{}} &&D_{6}^{\lambda}(x) = x^{6}\\ &&+\frac{-26624 \lambda^{8}-315136 \lambda^{7}-1452096 \lambda^{6}-3030464 \lambda^{5}-1350544 \lambda^{4}+6634848 \lambda^{3}+14325052 \lambda^{2}+11993936 \lambda+3814971) }{18 (\lambda+2) (2 \lambda+3)^{2} (512 \lambda^{5}+2944 \lambda^{4}+5208 \lambda^{3}+4 \lambda^{2}-8638 \lambda-6429)}x^{4}\\ &&-\frac{(-8192 \lambda^{7}-55552 \lambda^{6}-93408 \lambda^{5}+238480 \lambda^{4}+1249616 \lambda^{3}+2167224 \lambda^{2}+1773274 \lambda+577325) }{6 (\lambda+2) (2 \lambda+3)^{2} (512 \lambda^{5}+2944 \lambda^{4}+5208 \lambda^{3}+4 \lambda^{2}-8638 \lambda-6429)} x^{2}\\ &&+\frac{4 (2 \lambda+1)^{2} (80 \lambda^{3}+426 \lambda^{2}+753 \lambda+442)}{3 (2 \lambda+3)^{2} (512 \lambda^{5}+2944 \lambda^{4}+5208 \lambda^{3}+4 \lambda^{2}-8638 \lambda-6429)},\\\\ && D_{7}^{\lambda}(x) = x^{7}\\ &&+\frac{(-10240 \lambda^{8}-121088 \lambda^{7}-561408 \lambda^{6}-1198624 \lambda^{5}-656672 \lambda^{4}+2255736 \lambda^{3}+5154212 \lambda^{2}+4387984 \lambda+1408809) }{6 (\lambda+2) (2 \lambda+3)^{2} (512 \lambda^{5}+2944 \lambda^{4}+5208 \lambda^{3}+4 \lambda^{2}-8638 \lambda-6429)}x^{5}\\ &&+\frac{4096 \lambda^{8}+78848 \lambda^{7}+458688 \lambda^{6}+1074016 \lambda^{5}+295376 \lambda^{4}-3734688 \lambda^{3}-8130836 \lambda^{2}-7213054 \lambda-2452023}{18 (\lambda+2) (2 \lambda+3)^{2} (512 \lambda^{5}+2944 \lambda^{4}+5208 \lambda^{3}+4 \lambda^{2}-8638 \lambda-6429)}x^{3}\\ &&-\frac{2 (512 \lambda^{6}+3328 \lambda^{5}+7048 \lambda^{4}+828 \lambda^{3}-19042 \lambda^{2}-28747 \lambda-13742) }{3 (2 \lambda+3)^{2} (512 \lambda^{5}+2944 \lambda^{4}+5208 \lambda^{3}+4 \lambda^{2}-8638 \lambda-6429)}x,\\ && D_{8}^{\lambda}(x) = x^{8}\\ &&+\frac{-34816 \lambda^{8}-411392 \lambda^{7}-1916352 \lambda^{6}-4161280 \lambda^{5}-2589488 \lambda^{4}+6899568 \lambda^{3}+16600220 \lambda^{2}+14333968 \lambda+4637883}{18 (\lambda+2) (2 \lambda+3)^{2} (512 \lambda^{5}+2944 \lambda^{4}+5208 \lambda^{3}+4 \lambda^{2}-8638 \lambda-6429)}x^{6}\\ &&+\Big[\frac{253952 \lambda^{10}+4967424 \lambda^{9}+36636672 \lambda^{8}+134987136 \lambda^{7}+240904128 \lambda^{6}+28003872 \lambda^{5}-813980016 \lambda^{4}}{162 (\lambda+2) (2 \lambda+3)^{4} (512 \lambda^{5}+2944 \lambda^{4}+5208 \lambda^{3}+4 \lambda^{2}-8638 \lambda-6429)}\\ &&-\frac{1823644104 \lambda^{3}+1962244068 \lambda^{2}+1102018370 \lambda+259653399}{162 (\lambda+2) (2 \lambda+3)^{4} (512 \lambda^{5}+2944 \lambda^{4}+5208 \lambda^{3}+4 \lambda^{2}-8638 \lambda-6429)} \Big]x^{4}\\ &&-\frac{8 (6656 \lambda^{8}+61760 \lambda^{7}+214784 \lambda^{6}+252224 \lambda^{5}-437944 \lambda^{4}-1922704 \lambda^{3}-2812378 \lambda^{2}-1984129 \lambda-566938) }{27 (2 \lambda+3)^{4} (512 \lambda^{5}+2944 \lambda^{4}+5208 \lambda^{3}+4 \lambda^{2}-8638 \lambda-6429)}x^{2}\\ &&-\frac{16 (\lambda+2)^{3} (2 \lambda+1)^{2} (80 \lambda^{2}+266 \lambda+221)}{27 (2 \lambda+3)^{4} (512 \lambda^{5}+2944 \lambda^{4}+5208 \lambda^{3}+4 \lambda^{2}-8638 \lambda-6429)},\\\\ &&D_{9}^{\lambda}(x) = x^{9}\\ &&+\frac{-38912 \lambda^{8}-459520 \lambda^{7}-2148480 \lambda^{6}-4726688 \lambda^{5}-3208960 \lambda^{4}+7031928 \lambda^{3}+17737804 \lambda^{2}+15503984 \lambda+5049339 }{18 (\lambda+2) (2 \lambda+3)^{2} (512 \lambda^{5}+2944 \lambda^{4}+5208 \lambda^{3}+4 \lambda^{2}-8638 \lambda-6429)}x^{7}\\ &&+\Big[\frac{376832 \lambda^{10}+6912000 \lambda^{9}+49677312 \lambda^{8}+182130432 \lambda^{7}+333265728 \lambda^{6}+89989248 \lambda^{5}-952585632 \lambda^{4}}{162 (\lambda+2) (2 \lambda+3)^{4} (512 \lambda^{5}+2944 \lambda^{4}+5208 \lambda^{3}+4 \lambda^{2}-8638 \lambda-6429)}\\ &&-\frac{2231977416 \lambda^{3}+2437175184 \lambda^{2}+1380264434 \lambda+327276231}{162 (\lambda+2) (2 \lambda+3)^{4} (512 \lambda^{5}+2944 \lambda^{4}+5208 \lambda^{3}+4 \lambda^{2}-8638 \lambda-6429)} \Big]x^{5}\\ &&-\frac{2 (4096 \lambda^{9} + 166912 \lambda^{8} + 1357504 \lambda^{7} + 4568800 \lambda^{6} + 5470096 \lambda^{5} - 8399264 \lambda^{4} - 38672660 \lambda^{3} - 57223262 \lambda^{2} - 40687679 \lambda - 11707302) }{81 (2 \lambda+3)^{4} (512 \lambda^{5}+2944 \lambda^{4}+5208 \lambda^{3}+4 \lambda^{2}-8638 \lambda-6429)}x^{3}\\ &&+\frac{8 (\lambda+2)^{3} (512 \lambda^{5}+1664 \lambda^{4}-328 \lambda^{3}-8108 \lambda^{2}-13238 \lambda-7313)}{27 (2 \lambda+3)^{4} (512 \lambda^{5}+2944 \lambda^{4}+5208 \lambda^{3}+4 \lambda^{2}-8638 \lambda-6429)}x,\\\\ &&D_{10}^{\lambda}(x) = x^{10}\\ &&+\frac{(-14336 \lambda^{8}-169216 \lambda^{7}-793536 \lambda^{6}-1764032 \lambda^{5}-1276144 \lambda^{4}+2388096 \lambda^{3}+6291796 \lambda^{2}+5558000 \lambda+1820265)}{6 (\lambda+2) (2 \lambda+3)^{2} (512 \lambda^{5}+2944 \lambda^{4}+5208 \lambda^{3}+4 \lambda^{2}-8638 \lambda-6429)}x^{8}\\ &&+\Big[\frac{57344 \lambda^{1}0+ 1012736 \lambda^{9}+7164672 \lambda^{8}+26224384 \lambda^{7}+48985088 \lambda^{6}+18933696 \lambda^{5}-120883088 \lambda^{4}}{18 (\lambda+2) (2 \lambda+3)^{4} (512 \lambda^{5}+2944 \lambda^{4}+5208 \lambda^{3}+4 \lambda^{2}-8638 \lambda-6429)}\\ &&-\frac{296145544 \lambda^{3}+327852636 \lambda^{2}+187090450 \lambda+44609151}{18 (\lambda+2) (2 \lambda+3)^{4} (512 \lambda^{5}+2944 \lambda^{4}+5208 \lambda^{3}+4 \lambda^{2}-8638 \lambda-6429)} \Big]x^{6}\\ &&-\Big[\frac{4 (\lambda+2) (200704 \lambda^{1}0+ 5561856 \lambda^{9}+45776256 \lambda^{8}+174881280 \lambda^{7}+305832480 \lambda^{6}-27524064 \lambda^{5}) }{729 (2 \lambda+3)^{6} (512 \lambda^{5}+2944 \lambda^{4}+5208 \lambda^{3}+4 \lambda^{2}-8638 \lambda-6429)}\\ &&-\frac{1252067256 \lambda^{4}+2680180104 \lambda^{3}+2833083030 \lambda^{2}+1572495406 \lambda+366899565}{729 (2 \lambda+3)^{6} (512 \lambda^{5}+2944 \lambda^{4}+5208 \lambda^{3}+4 \lambda^{2}-8638 \lambda-6429)} \Big]x^{4}\\ &&+\frac{8 (\lambda+2)^{3} (45056 \lambda^{7}+308992 \lambda^{6}+680928 \lambda^{5}-126736 \lambda^{4}-3262160 \lambda^{3}-6273816 \lambda^{2}-5263402 \lambda-1726229) }{243 (2 \lambda+3)^{6} (512 \lambda^{5}+2944 \lambda^{4}+5208 \lambda^{3}+4 \lambda^{2}-8638 \lambda-6429)}x^{2}\\ &&+\frac{64 (\lambda+2)^{5} (2 \lambda+1)^{2} (80 \lambda^{2}+266 \lambda+221)}{243 (2 \lambda+3)^{6} (512 \lambda^{5}+2944 \lambda^{4}+5208 \lambda^{3}+4 \lambda^{2}-8638 \lambda-6429)}. \end{array} $$