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18 Orthogonal PolynomialsClassical Orthogonal Polynomials

§18.9 Recurrence Relations and Derivatives

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§18.9(i) Recurrence Relations

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Notes:
For (18.9.1), (18.9.2) see Szegö (1975, (4.5.1)). For Table 18.9.1, Rows 2, 9, 10, see Szegö (1975, (4.7.17), (5.1.10), (5.5.8)), respectively; Rows 3 and 4 are rewritings of elementary trigonometric identities in view of (18.5.1), (18.5.2); Row 7 is the special case \alpha=\beta=0 of (18.9.2).
Keywords:
Jacobi polynomials, classical orthogonal polynomials
Referenced by:
§18.5(iv), §3.5(vi)
Permalink:
http://dlmf.nist.gov/18.9.i
18.9.1p_{{n+1}}(x)=(A_{n}x+B_{n})p_{n}(x)-C_{n}p_{{n-1}}(x).
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Symbols:
n: nonnegative integer, p_{n}(x): polynomial of degree n, x: real variable, A_{n}: coefficient, B_{n}: coefficient and C_{n}: coefficient
Referenced by:
§18.21(ii), §18.9(i), Table 18.9.1
Permalink:
http://dlmf.nist.gov/18.9.E1
Encodings:
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For p_{n}(x)=\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right),

For the other classical OP’s see Table 18.9.1; compare also §18.2(iv).

Table 18.9.1: Classical OP’s: recurrence relations (18.9.1).
p_{n}(x) A_{n} B_{n} C_{n}
\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x\right) \frac{2(n+\lambda)}{n+1} 0 \frac{n+2\lambda-1}{n+1}
\mathop{T_{{n}}\/}\nolimits\!\left(x\right) 2-\delta_{{n,0}} 0 1
\mathop{U_{{n}}\/}\nolimits\!\left(x\right) 2 0 1
\mathop{T^{{*}}_{{n}}\/}\nolimits\!\left(x\right) 4-2\delta_{{n,0}} -2+\delta_{{n,0}} 1
\mathop{U^{{*}}_{{n}}\/}\nolimits\!\left(x\right) 4 −2 1
\mathop{P_{{n}}\/}\nolimits\!\left(x\right) \frac{2n+1}{n+1} 0 \frac{n}{n+1}
\mathop{P^{{*}}_{{n}}\/}\nolimits\!\left(x\right) \frac{4n+2}{n+1} -\frac{2n+1}{n+1} \frac{n}{n+1}
\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x\right) -\frac{1}{n+1} \frac{2n+\alpha+1}{n+1} \frac{n+\alpha}{n+1}
\mathop{H_{{n}}\/}\nolimits\!\left(x\right) 2 0 2n
\mathop{\mathit{He}_{{n}}\/}\nolimits\!\left(x\right) 1 0 n

§18.9(ii) Contiguous Relations in the Parameters and the Degree

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Notes:
For (18.9.3)–(18.9.5) see Rainville (1960, §138, (17), (16), (14)). For (18.9.6) see Szegö (1975, (4.5.4)). For (18.9.7) see Szegö (1975, (4.7.29)). For (18.9.8) substitute (18.7.15) or (18.7.16); the resulting formula is a special case of Rainville (1960, §138, (11)). (18.9.9)–(18.9.12) are rewritings of elementary trigonometric identities in view of (18.5.1)–(18.5.4). For (18.9.13) see Szegö (1975, (5.1.13)). For (18.9.14) see Szegö (1975, (5.1.14)).
Keywords:
classical orthogonal polynomials
Permalink:
http://dlmf.nist.gov/18.9.ii

Jacobi

18.9.3\mathop{P^{{(\alpha,\beta-1)}}_{{n}}\/}\nolimits\!\left(x\right)-\mathop{P^{{(% \alpha-1,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)=\mathop{P^{{(\alpha,\beta)% }}_{{n-1}}\/}\nolimits\!\left(x\right),
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Symbols:
\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right): Jacobi polynomial, n: nonnegative integer and x: real variable
A&S Ref:
22.7.20
Referenced by:
§18.9(ii)
Permalink:
http://dlmf.nist.gov/18.9.E3
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18.9.4(1-x)\mathop{P^{{(\alpha+1,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)+(1+x)% \mathop{P^{{(\alpha,\beta+1)}}_{{n}}\/}\nolimits\!\left(x\right)=2\!\mathop{P^% {{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right).
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Symbols:
\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right): Jacobi polynomial, n: nonnegative integer and x: real variable
A&S Ref:
22.7.17
Permalink:
http://dlmf.nist.gov/18.9.E4
Encodings:
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18.9.5(2n+\alpha+\beta+1)\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x% \right)=(n+\alpha+\beta+1)\mathop{P^{{(\alpha,\beta+1)}}_{{n}}\/}\nolimits\!% \left(x\right)+(n+\alpha)\mathop{P^{{(\alpha,\beta+1)}}_{{n-1}}\/}\nolimits\!% \left(x\right),
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Symbols:
\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right): Jacobi polynomial, n: nonnegative integer and x: real variable
A&S Ref:
22.7.19
Referenced by:
‣ §18.9(ii), §18.9(ii)
Permalink:
http://dlmf.nist.gov/18.9.E5
Encodings:
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18.9.6(n+\tfrac{1}{2}\alpha+\tfrac{1}{2}\beta+1)(1+x)\mathop{P^{{(\alpha,\beta+1)}}_% {{n}}\/}\nolimits\!\left(x\right)=(n+1)\mathop{P^{{(\alpha,\beta)}}_{{n+1}}\/}% \nolimits\!\left(x\right)+(n+\beta+1)\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}% \nolimits\!\left(x\right),
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Symbols:
\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right): Jacobi polynomial, n: nonnegative integer and x: real variable
A&S Ref:
22.7.16
Referenced by:
‣ §18.9(ii), §18.9(ii)
Permalink:
http://dlmf.nist.gov/18.9.E6
Encodings:
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and a similar pair to (18.9.5) and (18.9.6) by symmetry; compare the second row in Table 18.6.1.

Laguerre

§18.9(iii) Derivatives

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Notes:
For (18.9.15) see Szegö (1975, (4.21.7)). (18.9.16) is an immediate corollary of (18.5.5) and Table 18.5.1, Row 2. For (18.9.17) and (18.9.18) see Koornwinder (2006, §4). For (18.9.19) see Szegö (1975, (4.7.14)). (18.9.20) is an immediate corollary of (18.5.5) and Table 18.5.1, Row 3. (18.9.21) and (18.9.22) are rewritings of elementary trigonometric differentiation formulas. For (18.9.23) see Szegö (1975, (5.1.14)). (18.9.24) is an immediate corollary of (18.5.5) and Table 18.5.1, Row 9. For (18.9.25) see Szegö (1975, (5.5.10)). (18.9.26) is an immediate corollary of (18.5.5) and Table 18.5.1, Row 10.
Keywords:
classical orthogonal polynomials, derivatives
Permalink:
http://dlmf.nist.gov/18.9.iii
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