Orthogonal polynomials
This is a mild introduction to the theory of orthogonal polynomials written by a physics prac-titioner. Accordingly, the text does not claim rigor from the point of view of real analysis. Assuming that all functions dealt with are well-behaved, the arguments sketched below are, however, correct in the linear algebra sense. In spite of the lack of rigor, the text is organized into a set of theorems supplemented with proofs in order to highlight the main results and to separate them from auxiliary arguments.
Basic properties
We start by introducing an inner product on the vector space of real polynomials of a single real variable. For any two polynomials, P(x) and Q(x), we set
hP, Qi ≡
Z
P(x)Q(x)w(x) dx. (1)
Herew(x) is a positiveweight function. In order for the inner product to be well-defined,w(x) should decrease sufficiently fast forx→ ±∞, or the integration be done over a finite range only. All the information about the inner product can be encoded in a single moment functional,
L[P]≡
Z
P(x)w(x) dx, (2)
so that hP, Qi =L[P Q]. The real numbers µn ≡ L[xn] with non-negative integer n are called themoments associated with the weightw(x). By linearity, they determine uniquely the action of the functional (2) on any polynomial, and thus also the inner product (1).
Starting from the basis{1, x, x2, x3, . . .}, one can now, for given weight functionw(x), construct an orthogonal set of polynomials Pn(x) by using the Gram–Schmidt orthogonalization method. By construction,Pn(x),n = 0,1,2, . . ., is a polynomial of degreenand we havehPm, Pni= 0 for
m6=n. The normalization of the polynomials can be set, and the polynomials thus determined uniquely, either by fixing the leading coefficient κn, defined by
Pn(x) =κnxn+ polynomial of degree n−1 or less, (3)
or by fixing the squared norm λn with respect to the inner product (1),
hPm, Pni=λnδmn, (4)
together with a choice of sign for κn. Two special conventions for the normalization are often-times used, which will be distinguished by a dedicated notation:
• Monic orthogonal polynomialspn(x), for which κn = 1.
• Orthonormal polynomialsπn(x), for which λn = 1 andκn >0.
Theorem 1. The orthogonal polynomials Pn(x) satisfy the recurrence relation
xPn(x) =anPn+1(x) +bnPn(x) +cnPn−1(x), n≥1, (5)
where
an =
κn
κn+1
, cn=
γn
γn−1
, γn ≡ L[xnPn]. (6)
For monic orthogonal polynomials, an = 1, whereas for orthonormal polynomials, cn = an−1.
For even weight functions, w(x) =w(−x), the orthogonal polynomials have alternating parity, Pn(−x) = (−1)nPn(x), and bn = 0.
Proof. First of all, xPn(x) is a polynomial of degree n+ 1, and as such can be written as a linear combination of Pk(x) with 0 ≤k ≤ n+ 1. However, for k ≤ n−2 we havehxPn, Pki=
L[xPnPk] =hPn, xPki= 0, forPn(x) is by construction orthogonal to all polynomials of degree
n−1 or less. This confirms that onlyn−1≤k ≤n+ 1 contribute to the expansion ofxPn(x) in Pk(x). Using the orthogonality property (4), the coefficients of the expansion follow as
an =
L[xPnPn+1]
λn+1
, bn =
L[xP2 n]
λn
, cn =
L[xPnPn−1]
λn−1
. (7)
Next, we have, again by orthogonality of Pn(x) to polynomials of degree n−1 or less,
λn =hPn, Pni=hκnxn, Pni=κnL[xnPn] =κnγn. (8)
Finally, L[xPnPn+1] = L[κnxn+1Pn+1] = κnγn+1 by the same argument. Combining these two observations with Eq. (7) leads immediately to the result (6). The special cases of monic and orthonormal polynomials follow trivially; in the latter case, Eq. (8) implies γn= 1/κn.
The parity of the orthogonal polynomials together with bn = 0 for even weight functions is easily proven by induction. Indeed, P0(x) is constant and thus even, which by Eq. (7) implies
b0 = 0, and consequently P1(x) = ax0P0(x) is odd. This in turn implies b1 = 0. Suppose now that Pk(−x) = (−1)kPk(x) and bk = 0 hold for all k ≤ n with n ≥ 1. Then by Theorem 1,
Pn+1(x) = a1n[xPn(x) −cnPn−1(x)], which manifestly satisfies Pn+1(−x) = (−1)n+1Pn+1(x). Consequently, bn+1 = 0 by Eq. (7), which completes the induction step.
With the recurrence relation (5) at hand, we next derive an important auxiliary result.
Theorem 2 (Christoffel–Darboux). The orthogonal polynomials Pn(x) satisfy the identity n
X
k=0
Pk(x)Pk(y)
λk
= an
λn
Pn+1(x)Pn(y)−Pn(x)Pn+1(y)
x−y , (9)
and its confluent form
n X
k=0
[Pk(x)]2
λk
= an
λn
[Pn+10 (x)Pn(x)−Pn0(x)Pn+1(x)]. (10)
Proof. The recurrence relation (5) gives
(x−y)Pk(x)Pk(y)
= [akPk+1(x) +bkPk(x) +ckPk−1(x)]Pk(y)−Pk(x)[akPk+1(y) +bkPk(y) +ckPk−1(y)]
= [akPk+1(x)Pk(y)−ckPk(x)Pk−1(y)]−[akPk(x)Pk+1(y)−ckPk−1(x)Pk(y)].
Next we need to relate the coefficients an and cn. By Theorem 1 and Eq. (8), we get
cn=
γn
γn−1 = λn
κn
κn−1
λn−1
=an−1
λn
λn−1
. (12)
Dividing Eq. (11) by λk then leads to
(x−y)Pk(x)Pk(y)
λk
=
ak
λk
Pk+1(x)Pk(y)−
ak−1
λk−1
Pk(x)Pk−1(y)
−
ak
λk
Pk(x)Pk+1(y)−
ak−1
λk−1
Pk−1(x)Pk(y)
.
(13)
Summing this over k= 0, . . . , nand using P−1(x)≡0 as the initial condition for the recursion proves Eq. (9). The confluent form (10) follows straightforwardly by taking the limity→x.
We are now ready to analyze the basic properties of Pn(x) as functions of a single real variable. We start with the elementary observation that all the orthogonal polynomials have only real roots with single multiplicity.
Theorem 3. The polynomial Pn(x) has n different real roots, all with single multiplicity.
Proof. Suppose that the polynomial has only m < n real roots x1, . . . , xm with odd (whether single or higher) multiplicity. Then the polynomial
Q(x)≡(x−x1)· · ·(x−xm)Pn(x)≡Q˜(x)Pn(x) (14)
has only real roots with even multiplicity or complex-conjugate pairs of complex roots. As such, it has a constant sign over the whole real axis and thus necessarily L[Q] 6= 0. But at the same time, we can interpret L[Q] as hQ, P˜ ni. By assumption, ˜Q(x) has degree less thann and thus should be orthogonal to Pn(x), leading to a contradiction. We conclude that Pn(x) has at least n real roots with odd multiplicity, which by the fundamental theorem of algebra immediately implies that it has exactlyn real roots, all of them having single multiplicity.
One can, however, make a much stronger statement, relating roots of the different polynomials.
Theorem 4. The roots of Pn(x) and Pn+1(x) alternate. That is, if x(n)1 < · · · < x(n)n are the
ordered roots ofPn(x)andx(n+1)1 <· · ·< x(n+1)n+1 the ordered roots of Pn+1(x), then the following inequality holds,
x(n+1)1 < x(n)1 < x(n+1)2 < x(n)2 < x3(n+1) <· · ·< x(n+1)n < x(n)n < x(n+1)n+1 . (15)
Proof. Evaluating Eq. (10) on a selected root of Pn+1(x), say x (n+1) m , gives n
X
k=0 [Pk(x
(n+1) m )]2
λk
= an
λn
Pn+10 (x(n+1)m )Pn(x(n+1)m ). (16)
The left-hand side is non-negative, and sinceP0(x) is a nonzero constant, it is necessarily strictly positive. This means that the product Pn+10 (x(n+1)m )Pn(x
(n+1)
m ) has the same sign (depending on the sign of an) for all the roots x
(n+1)
Matrix representation
As mentioned above, the polynomials Pn(x) are determined uniquely by their orthogonality property and by fixing the value of one of the coefficients γn, κn, λn. (In case of fixing λn, a definite sign ofκnhas to be chosen in addition.) This means that there must be two independent constraints, relating γn, κn, λn. One of them is provided by Eq. (8). To find another, we will now relate the coefficients to the properties of the weight function w(x).
We start by integrating the definition of γn in Eq. (6) and the orthogonality property ofPn(x) into a single compact expression,
L[xmPn] =δmnγn for all m≤n. (17)
This can be recast in terms of the moments µn by expanding Pn(x) in the monomial basis,
Pn(x)≡ n X
k=0
αn,kxk =⇒ n X
k=0
αn,kµm+k =δmnγn. (18)
Further rewriting this in a matrix form then leads straightforwardly to the following theorem.
Theorem 5. Let ∆n be the (n+ 1)×(n+ 1) matrix with (∆n)k`≡µk+`, wherek, `= 0, . . . , n.
Then the orthogonal polynomial Pn(x) can be represented by the determinant expression
Pn(x) =
γn
|∆n|
µ0 µ1 · · · µn
µ1 µ2 · · · µn+1
..
. ... . .. ... µn−1 µn · · · µ2n−1
1 x · · · xn
. (19)
As a corollary, the leading coefficient of Pn(x) is given by
κn=γn
|∆n−1|
|∆n|
. (20)
Proof. Eq. (18) represents a set of linear equations forαn,k that can be cast in a matrix form as ∆n~αn =~Γn, where α~n≡ (αn,0, . . . , αn,n)T and ~Γn ≡(0, . . . ,0, γn)T. This is solved by inverting the matrix ∆n, and the polynomial Pn(x) can then be formally written as
Pn(x) = (1, x, . . . , xn)α~n= (1, x, . . . , xn)∆−n1~Γn. (21)
Upon working out the inverse ∆−n1 using Cramer’s rule, and expanding Eq. (19) in powers of x
using the Laplace expansion method, the two expressions for Pn(x) are seen to coincide. The corollary relating the leading coefficientκntoγnand the determinants|∆n|follows immediately as a byproduct of the Laplace expansion. Note that all the above expressions involving the determinant and inverse of ∆n are well-defined, since the matrix ∆n is positive-definite as a consequence of the positivity of the weight function w(x). Indeed, for an arbitrary nonzero
polynomial of degreen, Q(x)≡
n P k=0
αkxk, we have
0<hQ, Qi= n X
k,`=0
αkα`µk+` =~αT∆nα.~ (22)
By combining the relation (20) with Eq. (8), we obtain a direct relation between κn and λn,
λn =κ2n
|∆n|
|∆n−1|
. (23)
This gives an explicit expression for the norm of monic polynomials, or the leading coefficient of the orthonormal polynomials, in terms of the moments of the weight function w(x).
Apart from Eq. (19), there is another determinant representation for the orthogonal polynomials
Pn(x), which sheds some more light on the properties of their roots.
Theorem 6. The orthogonal polynomial Pn(x) can be represented as
Pn(x) = κn|x11−Mn|, or equivalently pn(x) =|x11−Mn|, (24)
where Mn is a tridiagonal matrix built of the coefficients of the recurrence relation (5),
Mn≡
bn−1 c˜n−1 0 · · · 0
˜
cn−1 bn−2 c˜n−2 . .. ... 0
0 c˜n−2 bn−3 . .. ... ...
..
. . .. . .. ... ˜c2 0
..
. . .. . .. c˜2 b1 c˜1 0 0 · · · 0 ˜c1 b0
, c˜n≡
κn−1
κn s
λn
λn−1 =cn
r
λn−1
λn
. (25)
Proof. Denote the right-hand side of Eq. (24) asRn(x). We setR0(x)≡κ0. Moreover, we have
R1(x) = κ1(x−b0), which is seen to coincide withP1(x) upon setting n= 0 and P−1(x)≡0 in Eq. (5). Using the Laplace expansion, we can now derive a recurrence formula for Rn(x),
anRn+1(x) =κn|x11−Mn+1|=κn
(x−bn)|x11−Mn| −˜c2n|x11−Mn−1|
= (x−bn)Rn(x)−
κn−1
κn
λn
λn−1
κn−1|x11−Mn−1|= (x−bn)Rn(x)−cnRn−1(x), (26)
where we used Eq. (12) in the last step. A glance at Eq. (5) shows that the polynomialsRn(x) satisfy the same recurrence relation asPn(x). Since also the initial conditions for the recurrence,
n = 0,1, agree, the two sets of polynomials must be identical.
Theorem 6 immediately leads to the observation that the roots of Pn(x) correspond to the eigenvalues of the matrix Mn. Since Mn is a real symmetric matrix, its eigenvectors form a complete orthogonal system of vectors onRn. This is the subject of the following two theorems.
Theorem 7. The eigenvalues of the matrix Mn give the roots of the polynomial Pn(x). The
eigenvector corresponding to the root x(n)m ≡xm is, up to an arbitrary normalization factor,
~ vm =
Pn−1(xm)/ p
λn−1
Pn−2(xm)/ p
λn−2
.. . P0(xm)/√λ0
=
πn−1(xm) sgnκn−1
πn−2(xm) sgnκn−2
.. . π0(xm) sgnκ0
Proof. The k-th element of the vector Mn~vm, wherek = 1, . . . , n, reads1
(Mn~vm)k = ˜cn−k+1
Pn−k+1(xm) p
λn−k+1
+bn−k
Pn−k(xm) p
λn−k
+ ˜cn−k
Pn−k−1(xm) p
λn−k−1
= p1
λn−k
an−kPn−k+1(xm) +bn−kPn−k(xm) +cn−kPn−k−1(xm)
=xm
Pn−k(xm) p
λn−k
=xm(~vm)k,
(28)
where we used the relation for ˜cn in Eq. (25), Eq. (6), and finally the recurrence formula (5). This proves both that~vm is an eigenvector and that xm is the corresponding eigenvalue.
Theorem 8. Let x`, xm be two roots of Pn(x). Then the following relations hold,
n−1 X
k=0
Pk(x`)Pk(xm)
λk
=ξ`δ`m, `, m= 1, . . . , n, (29)
n X
m=1
Pk(xm)P`(xm)
ξm
=λkδk`, k, `= 0, . . . , n−1, (30)
where
ξm ≡
an−1
λn−1
Pn0(xm)Pn−1(xm). (31)
Proof. The validity of Eq. (29) for `6= m merely expresses the orthogonality of the vectors ~v` and~vm from the previous theorem. All that needs to be done to prove Eq. (29) is therefore to evaluate the norm of~vm. The given result forξm follows immediately from the confluent form of the Christoffel–Darboux theorem, Eq. (10), by replacing n→n−1 and x→xm therein. Note that the orthogonality of two different eigenvectors~v`,~vm also follows directly from Eq. (9) by setting x→x` and y →xm therein.
Eq. (29) shows that ~um ≡~vm/
√
ξm defines a set of n orthonormal vectors in Rn. As such, it satisfies the completeness relation
δk` = n X
m=1
(~um)k(~um)` = n X
m=1 1
ξm
Pn−k(xm)
λn−k
Pn−`(xm)
λn−`
, (32)
which is equivalent to Eq. (30).
Gauss quadrature
An important application of the theory of orthogonal polynomials is the method of Gauss quadrature, giving an approximation to the integral of a given function using its values at a finite set of points. We start with a reminder of the Lagrange interpolation method.
Theorem 9 (Lagrange interpolation). Let x1 <· · ·< xn be a set of n distinct points and f(x)
a given function. Then there is a unique polynomial P(x) of degree n −1 or less such that P(xk) = f(xk) for all k = 1, . . . , n. If f(x) itself is a polynomial of degree n−1 or less, then P(x) =f(x) for any x.
1This expression is formally also valid fork= 1 sinceP
Proof. DefineL(x)≡(x−x1)· · ·(x−xn) and construct the polynomials of degreen−1,
Lk(x)≡
L(x)
(x−xk)L0(xk). (33)
It is easy to verify that these polynomials satisfyLk(x`) =δk`. It then follows at once that the polynomial
P(x)≡
n X
k=1
f(xk)Lk(x) (34)
has the desired property, P(xk) =f(xk) for allk = 1, . . . , n. In casef(x) itself is a polynomial, then P(x)−f(x) is a polynomial of degree at most n−1, which however has n distinct roots,
x=xk,k = 1, . . . , n. By the fundamental theorem of algebra, this is only possible ifP(x)−f(x) is identically zero.
The approximation of a given function f(x) by a polynomial P(x) can now be used to carry out approximate integration, or quadrature, off(x), since integration of polynomials is trivial. Note that the polynomialsLk(x) only depend on the choice of the pointsx1, . . . , xn and not on the unknown function f(x), and hence the integration of P(x) can be performed a priori; for given f(x) one then just needs to supply the values f(xk). Since the Lagrange interpolation is exact for polynomials of degree up to n−1, we thus have a quadrature method that is exact for the same class of polynomials. Gauss improved substantially on the accuracy of numerical integration by showing that with a suitable choice of the points x1, . . . , xn, the integration can be made exact for polynomials of degree up to 2n−1. The trick is to use our knowledge of orthogonal polynomials to reduce integration of a polynomial of degree 2n−1 to that of an auxiliary polynomial of degree n−1, and then use Theorem 9.
Theorem 10 (Gauss quadrature). Let {Pn(x)}∞n=0 be a set of orthogonal polynomials with
respect to the weight function w(x) and let xm, m = 1, . . . , n be the roots of Pn(x). Then for
an arbitrary polynomial f(x) of degree up to 2n−1,
Z
f(x)w(x) dx= n X
m=1
wmf(xm), (35)
where wm is a set of positive weights, independent of the function f(x).
Proof. We start by decomposing the polynomialf(x) of degree 2n−1 in terms of two polyno-mials, g(x) and h(x), of degree n−1,
f(x) =g(x)Pn(x) +h(x); (36)
this merely says that h(x) is the remainder upon the division of f(x) byPn(x). The orthogo-nality of Pn(x) to all polynomials of degree less than n then implies that
Z
f(x)w(x) dx=hg, Pni+ Z
h(x)w(x) dx= Z
h(x)w(x) dx
= Z n
X
m=1
h(xm)Lm(x)w(x) dx= n X
m=1
wmf(xm),
where
wm ≡ Z
Lm(x)w(x) dx= Z
Pn(x) (x−xm)Pn0(xm)
w(x) dx. (38)
It the third step, we used the Lagrange interpolation formula (34) with pn(x) = Pn(x)/κn in place of L(x), choosing of course the n points xm as the roots of Pn(x). This guarantees that
f(xm) = h(xm), as used in the last step. It remains to prove that the weights wm as defined by Eq. (38) are positive. To that end, one simply has to apply the already proven quadrature formula (37) to the function f(x) = [L`(x)]2 and use the fact thatL`(xm) = δ`m,
w` = n X
m=1
wm[L`(xm)]2 = Z
[L`(x)]2w(x) dx. (39)
This proves thatw` >0 as a consequence of the positivity of w(x) itself.
There is another expression for the Gaussian weights wm which also makes their positivity manifest, and moreover connects them directly to the roots of the polynomials Pn(x).
Theorem 11. The Gaussian weights wm are related to the squared norms ξm of Theorem 8 by
wm = 1
ξm
. (40)
Proof. Take the Christoffel–Darboux identity (9), replacing n → n −1 and y → xm therein, where xm is a root ofPn(x). This gives
n−1 X
k=0
Pk(x)Pk(xm)
λk
= an−1
λn−1
Pn(x)Pn−1(xm)
x−xm
. (41)
Now integrate with the weight w(x). The right-hand side is related to the Gaussian weight
wm via Eq. (38), whereas on the left-hand side, only the k = 0 term survives thanks to the orthogonality of Pn(x) to all polynomials of lower degree. Finally, using the fact that P0(x) is a constant, the whole equation reduces to
1 = Z P2
0
λ0
w(x) dx= an−1
λn−1
wmPn0(xm)Pn−1(xm) =wmξm, (42)
where in the last step, we used Eq. (31).
Note that with wm = 1/ξm and the Gauss quadrature theorem 10 at hand, the left-hand side of Eq. (30) becomeshPk, P`i. The completeness of the set of eigenvectors~vm is then equivalent to the orthogonality of the original polynomials Pn(x).
Before concluding the section, we finally mention one interesting application of Gauss quadra-ture, which sharpens somewhat the result of Theorem 4 on the ordering of the roots of the polynomials Pn(x).
Theorem 12. For any two integersm, nsuch that 0≤m < n, between any two roots of Pm(x),
Proof. It is sufficient to consider two neighboring roots of Pm(x), x (m)
k and x (m)
k+1, ordered for definiteness so thatx(m)k < x(m)k+1. Now construct an auxiliary polynomial of degree m−2,
Q(x)≡ Pm(x)
(x−x(m)k )(x−x(m)k+1). (43)
The polynomial Q(x)Pm(x) has degree 2m−2<2n−1, and its integral can thus be evaluated using the Gauss quadrature formula (35),
hQ, Pmi=
Z [P m(x)]2
(x−x(m)k )(x−x(m)k+1)w(x) dx= n X
`=0
w`
[Pm(x (n) ` )]2
(x(n)` −x(m)k )(x(n)` −x(m)k+1), (44)
where w` are the Gaussian weights for Pn(x). Should none of the roots x (n)
` satisfy x (m) k <
x(n)` < x(m)k+1, the last expression would be manifestly positive in contradiction to the fact that
hQ, Pmi= 0 by the orthogonality property of Pm. This concludes the proof.
Examples
In this final section, we list some concrete examples of orthogonal polynomials that find frequent use in physics. Unlike in the rest of the text, it is not the intention here to derive or prove all the results. We rather want to provide a potentially useful summary of the basic properties of these classic series of orthogonal polynomials. Remarkably, some of the properties of the orthogonal polynomials listed below can be understood on a common ground. This applies in particular to the explicit expression for Pn(x), given by the so-called Rodrigues formula,
Pn(x)∝ 1 w(x)
dn dxn
w(x)[ρ(x)]n , (45)
where ρ(x) is a polynomial independent of n of degree two or less. In addition, all the below-discussed orthogonal polynomials satisfy a differential equation of the Sturm–Liouville type,
w(x)ρ(x)y0(x)0
−σnw(x)y(x) = 0, (46)
and thus represent eigenvectors of certain self-adjoint differential operator with eigenvalue σn.
Jacobi polynomials
The Jacobi polynomials Pn(α,β)(x) are orthogonal on the interval (−1,+1) with the weight function w(x) = (1−x)α(1 +x)β, whereα, β >−1. (The special case withα =β is known as
Gegenbauer or ultraspherical polynomials.) They can be expressed explicitly by the Rodrigues formula (45), which in this case reads
Pn(α,β)(x) = (−1) n
2nn! 1
w(x) dn dxn
w(x)(1−x2)n
= 1 2n
n X
k=0
n+α n−k
n+β k
(x−1)k(x+ 1)n−k.
The leading coefficient κn and the squared norm λn of the Jacobi polynomials are given by
κn = 1 2n
2n+α+β n
, λn =
2α+β+1Γ(n+α+ 1)Γ(n+β+ 1)
(2n+α+β+ 1)n!Γ(n+α+β+ 1). (48)
Furthermore, the Jacobi polynomials satisfy a second-order linear differential equation of the type (46) with ρ(x) = 1−x2 and σ
n=−n(n+α+β+ 1), or more explicitly
(1−x2)y00(x) + [β−α−(α+β+ 2)x]y0(x) +n(n+α+β+ 1)y(x) = 0. (49)
The master recurrence relation (5) takes the explicit, if somewhat lengthy, form
2n(n+α+β)(2n+α+β−2)Pn(α,β)(x)
= (2n+α+β−1)
(2n+α+β)(2n+α+β−2)x+α2−β2
Pn(α,β)−1 (x)
−2(n+α−1)(n+β−1)(2n+α+β)Pn(α,β)−2 (x).
(50)
We also have simpler, derivative-type recurrence relations,
(2n+α+β)(1−x2)dP (α,β) n (x)
dx =n
α−β−(2n+α+β)x]Pn(α,β)(x)
+ 2(n+α)(n+β)Pn(α,β)−1 (x),
(51)
and
dPn(α,β)(x) dx =
1
2(n+α+β+ 1)P
(α+1,β+1)
n−1 (x). (52)
Finally, the Jacobi polynomials can be obtained from the generating function
2α+β
R(1 +R−z)α(1 +R+z)β =
∞
X
n=0
Pn(α,β)(x)zn, where R≡√1−2xz+z2. (53)
Legendre polynomials
This is a special case of Jacobi polynomials whereα =β = 0, and thus w(x) = 1. Just like the Jacobi polynomials, the Legendre polynomials form an orthogonal set on the interval (−1,+1). They are given explicitly by the Rodrigues formula (45),
Pn(x) = 1 2nn!
dn dxn(x
2−1)n. (54)
The leading coefficient κn and the squared norm λn are given by
κn= 1 2n
2n
n
, λn = 2
2n+ 1. (55)
The Legendre differential equation, following from Eq. (46) by setting ρ(x) = 1 − x2 and
σn=−n(n+ 1), takes the form
The different Legendre polynomials can be related to each other by means of the recurrence relations
(n+ 1)Pn+1(x) = (2n+ 1)xPn(x)−nPn−1(x), (57) (1−x2)Pn0(x) = −nxPn(x) +nPn−1(x). (58)
Finally, the generating function for the Legendre polynomials reads
1
√
1−2xz+z2 =
∞
X
n=0
Pn(x)zn. (59)
Chebyshev polynomials
There are two classes of Chebyshev polynomials, both of which are special cases of the Jacobi polynomials. The Chebyshev polynomials of the first kind,Tn(x), correspond to the choiceα=
β = −1
2 and thus w(x) = 1/
√
1−x2. Conventionally, the normalization is chosen differently from the Jacobi polynomials,
Tn(x) = 22n
2n n
−1
P(−
1 2,−
1 2)
n (x). (60)
This allows a simple trigonometric representation of the polynomials,
Tn(cosθ) = cosnθ. (61)
The Chebyshev polynomials of the first kind satisfy the differential equation
(1−x2)y00(x)−xy0(x) +n2y(x) = 0, (62)
and the recurrence relation
Tn+1(x) = 2xTn(x)−Tn−1(x). (63)
They can be obtained from the generating function
1−xz
1−2xz+z2 =
∞
X
n=0
Tn(x)zn. (64)
The Chebyshev polynomials of the second kind, Un(x), correspond to the Jacobi polynomials with α=β = 12, hence w(x) =√1−x2. The normalization is chosen so that
Un(x) = 22n
2n+ 1
n+ 1 −1
P(
1 2,
1 2)
n (x), Un(cosθ) =
sin(n+ 1)θ
sinθ . (65)
They satisfy a different differential equation,
(1−x2)y00(x)−3xy0(x) +n(n+ 2)y(x) = 0, (66)
but the same recurrence relation as the polynomials of the first kind,
They can be obtained from the generating function
1
1−2xz+z2 =
∞
X
n=0
Un(x)zn. (68)
There are numerous relations connecting the two classes of Chebyshev polynomials. Let us mention at least the recurrence-like identities
Tn(x) =Un(x)−xUn−1(x), Tn0(x) =nUn−1(x). (69)
There is also a remarkable addition formula, connecting the two classes of Chebyshev polyno-mials to the Legendre polynopolyno-mials,
Un(x) = n X
k=0
Tk(x)xn−k = n X
k=0
Pk(x)Pn−k(x). (70)
Hermite polynomials
The Hermite polynomials Hn(x) are orthogonal on the interval (−∞,+∞) with respect to the weight functionw(x) = e−x2. They can be expressed by means of the Rodrigues formula (45),
Hn(x) = (−1)nex
2 dn
dxne
−x2
=n! [n/2] X
k=0
(−1)k(2x)n−2k
k!(n−2k)! . (71)
The leading coefficient κn and the squared norm λn of the Hermite polynomials are equal to
κn= 2n, λn = 2nn!
√
π. (72)
The Hermite differential equation corresponds to Eq. (46) with ρ(x) = 1 and σn =−2n,
y00(x)−2xy0(x) + 2ny(x) = 0. (73)
Similarly to the Jacobi polynomials, we have a non-derivative three-term and a derivative two-term recurrence relation,
Hn+1(x) = 2xHn(x)−2nHn−1(x), (74)
Hn0(x) = 2nHn−1(x). (75)
The Hermite polynomials can be obtained from the simple generating function,
e2xz−z2 =
∞
X
n=0
Hn(x)
zn
Laguerre polynomials
The (generalized) Laguerre polynomialsLα
n(x) are orthogonal on the interval [0,+∞) with the weight function w(x) = xαe−x, where α > −1. They are given explicitly by the Rodrigues formula (45) withρ(x) = x,
Lαn(x) = 1
n! 1
w(x) dn dxn
w(x)xn= n X
k=0
n+α n−k
(−x)k
k! . (77)
The leading coefficient κn and the squared norm λn are given respectively by
κn = (−1)n
n! , λn=
Γ(n+α+ 1)
n! . (78)
The Laguerre polynomials satisfy a differential equation of the type (46) with σn=−n,
xy00(x) + (α+ 1−x)y0(x) +ny(x) = 0. (79)
The master recurrence formula (5) takes the form
(n+ 1)Lαn+1(x) + (x−2n−α−1)Lαn(x) + (n+α)Lαn−1(x) = 0. (80)
A number of simpler relations exists though, which involve derivatives of the Laguerre polyno-mials, for instance
xdL
α n(x) dx =nL
α
n(x)−(n+α)L α
n−1(x), (81) or
dLαn(x)
dx =−L
α+1
n−1(x) (82)
and its generalization,
dkLα n(x)
dxk = (−1) kLα+k
n−k(x) if k≤n and 0 otherwise. (83)
A compact expression for the derivative of the Laguerre polynomial with respect to the param-eter α also exists,
dLα n(x) dα =
n−1 X
k=0
Lα k(x)
n−k. (84)
Out of the many further remarkable properties of the Laguerre polynomials we list at least the addition formula,
Lα+β+1n (x+y) = n X
k=0
Lαk(x)Lβn−k(y). (85)
A number of the above identities can be proven straightforwardly with the help of the generating function,
1
(1−z)α+1 exp
− xz
1−z
=
∞
X
n=0
