For (18.9.1), (18.9.2) see
Szegő (1975, (4.5.1)).
For Table 18.9.1, Rows 2, 9, 10, see
Szegő (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 of
(18.9.2).
Initial values and for the general recurrence relation have been added
in terms of the coefficients and , and these coefficients have been explicitly
specified for Jacobi polynomials in this subsection.
For the monic versions of the classical OP’s the recurrence
coefficients and (there written as and ,
respectively) are given in §3.5(vi). They imply the recurrence
coefficients for the orthonormal versions of the classical OP’s as well,
see again §3.5(vi).
§18.9(ii) Contiguous Relations in the Parameters and the Degree
For (18.9.3)–(18.9.5) see
Rainville (1960, §138, (17), (16), (14)).
For (18.9.6) see Szegő (1975, (4.5.4)).
For (18.9.7) see Szegő (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 Szegő (1975, (5.1.13)).
For (18.9.14) see Szegő (1975, (5.1.14)).
The DLMF now adopts the definitions for the Chebyshev polynomials
of the third and fourth kinds ,
used in Mason and Handscomb (2003). Therefore , ,
having been interchanged, on the left-hand side we replaced
with .
For further details see Errata.
The DLMF now adopts the definitions for the Chebyshev polynomials
of the third and fourth kinds ,
used in Mason and Handscomb (2003). Therefore , ,
having been interchanged, on the right-hand side we replaced
with .
For further details see Errata.
For (18.9.15) see Szegő (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 Szegő (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 Szegő (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 Szegő (1975, (5.5.10)).
(18.9.26) is an immediate corollary of
(18.5.5) and Table 18.5.1, Row 10.
Further -th derivative formulas relating two different Jacobi polynomials
can be obtained from §15.5(i) by substitution of
(18.5.7).
Formula (18.9.15) is degree lowering, while it raises the
parameters. Formula (18.9.16) is degree raising,
while it lowers
the parameters. The following three
formulas change the degree but preserve the parameters, see
(18.2.42)–(18.2.44) for similar formulas
for more general OP’s.