aa r X i v : . [ m a t h . C A ] A p r CLASSICAL STURMIAN SEQUENCES
ALEXEI ZHEDANOV
Abstract.
The Sturm sequence is generated by a pair of polynomials P ( x ) and P ′ ( x ) , where P ( x ) is assumed to have simple real roots. Euclidean algorithm generates then a finite sequenceof polynomials orthogonal on the grid x s of roots of the polynomial P ( x ) . This algorithm can beexploited in order to find the number of roots of the polynomial P ( x ) inside a given interval. Weconsider the ”inverse” problem: what is the explicit system of orthogonal polynomials generatedby the prescribed grid x s of ”classical” type. The main results are the following. The genericlinear grid generates a special case of the Hahn polynomials. The quadratic grids x s = x ( s + ) and x s = s ( s + ) correspond to two special cases of the Racah polynomials. The generic exponentialgrid is related to a special case of the q-Hahn polynomials. Finally, we show that two specialtrigonometric grids are related to the Chebyshev polynomials of the first and second kind. Introduction
The classical Sturm algorithm [18] allows to find the number of real roots of the polynomial P ( x ) inside an interval. The main idea of the algorithm is application the Euclidean algorithm whichgenerates a sequence of polynomials of decreasing degrees starting from the pair of polynomials P ( x ) and P ′ ( x ) where P ′ ( x ) is the derivative of P ( x ) . Assume for simplicity that all roots of thepolynomial P ( x ) are real and simple. Then all the roots of the polynomial P ′ ( x ) are also realand simple and interlace the roots of the polynomial P ( x ) . This means that every root of thepolynomial P ′ ( x ) is located between two neighbor roots of the polynomial P ( x ) .In more general situation, assume that P N + ( x ) is a monic polynomial with N + x s . These zeros are ordered by increasing x < x < x < . . . x N . (1.1)Assume also the monic polynomial P N ( x ) of degree N is arbitrary with the only condition thatits simple real zeros interlace zeros of P N + ( x ) . Dividing P N + ( x ) by P N ( x ) we get P N + ( x ) = ( x − b N ) P N ( x ) − u N P N − ( x ) , (1.2)where P N − ( x ) is a monic polynomial of degree N − P N ( x ) . Thecoefficient u N is strictly positive u N >
0. This process can be continued which yields the chain ofmonic polynomials P N − ( x ) , P N − ( x ) , . . . , P =
1. They satisfy the recurrence relation P n + ( x ) = ( x − b n ) P n ( x ) − u n P n − ( x ) , n = , , . . . , N (1.3)with strictly positive coefficients u n > u n >
0) is equivalentto the statement that polynomials P n ( x ) , n = , , . . . , N are orthogonal N ∑ s = w s P n ( x s ) P m ( x s ) = h n δ nm (1.4)where the normalization coefficient h n = u u . . . u n , h = , (1.5)and where the positive weights w s > w s = h N P ′ N + ( x s ) P N ( x s ) . (1.6)The weights w s are normalized N ∑ s = w s = . (1.7) Thus starting with any pair P N + ( x ) , P N ( x ) of monic polynomials with simple interlacing zeros,one can reconstruct a family of polynomials P n ( x ) , n = , , . . . , N − x s of the polynomial P N + ( x ) with positive weight function. Hence, the Jacobi matrix J = ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝ b u b u b ⋱ ⋱ . . . u N − b N − . . . u N b N ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠ . can uniquely be associated with the pair P N + ( x ) , P N ( x ) [4].The mirror-dual Jacobi matrix J ∗ has the entries b ∗ n = b N − n , u ∗ n = u N + − n , n = , , . . . N (1.8)(it is assumed in (1.8) that u = u N + = J ∗ satisfies therelation J ∗ = RJ T R, (1.9)where J T is transposed Jacobi matrix and where R is the reflection matrix [7] R = ⎛⎜⎜⎜⎝ . . . . . . . . . . . . . . . . . . . . . . . . ⎞⎟⎟⎟⎠ . Corresponding mirror-dual orthogonal polynomials P ∗ n ( x ) are uniquely defined by the recurrencerelation P ∗ n + ( x ) = ( x − b ∗ n ) P ∗ n ( x ) − u ∗ n P ∗ n − ( x ) (1.10)and by the initial conditions P ∗ = , P ∗ ( x ) = x − b ∗ = x − b N . (1.11)The mirror-dual polynomials satisfy the orthogonality relation N ∑ s = w ∗ s P ∗ n ( x s ) P ∗ m ( x s ) = h ∗ n δ nm , (1.12)where the (normalized) the dual weights w ∗ s are w ∗ s = P N ( x s ) P ′ N + ( x s ) . (1.13)Comparing (1.13) with (1.6) we arrive at the important duality relation between the weights [2],[3], [17] w s w ∗ s = h N ( P ′ N + ( x s )) . (1.14)So far, the polynomial P N ( x ) was an arbitrary monic polynomial of degree N with the onlycondition of interlacing zeros with respect to P N + ( x ) . The Jacobi matrices J , J ∗ and orthogonalpolynomials P n ( x ) , P ∗ n ( x ) are in one-to-one correspondence with the choice of the polynomials P N ( x ) , P N + ( x ) . Consider now the special Sturmian case when P N ( x ) = ( N + ) − P ′ N + ( x ) . Thenfrom (1.13) we get w ∗ s = N + . (1.15)Thus for the Sturmian case all the dual weights w ∗ s are the same (i.e. they do not depend on s ).But the weights w ∗ s and the grid x s define uniquely the matrices J and J ∗ and hence - the pair ofpolynomials P N + ( x ) , P N ( x ) . It is natural to call the polynomials with constant weights (1.15) the Legendre-type polynomials. Indeed, the Legendre polynomials are orthogonal with the constantweight 1 / [ − , ] [8].We thus have LASSICAL STURMIAN SEQUENCES 3
Proposition 1.
The Sturmian pair P N + ( s ) and P N ( x ) = ( N + ) − P ′ N + ( x ) corresponds to mirror-dual orthogonal polynomials P ∗ n ( x ) of Legendre type. Conversely, any finite system of orthogonalpolynomials P ∗ n ( x ) , n = , , . . . , N with the weights w ∗ s = ( N + ) − , corresponds to the Sturmianpair P N + ( x ) and P N ( x ) = ( N + ) − P ′ N + ( x ) . We can therefore reduce the problem of classification of orthogonal polynomials P n ( x ) corre-sponding to the Sturmian pair P N + ( s ) and P N ( x ) = ( N + ) − P ′ N + ( x ) , to the problem of findingpolynomials P ∗ n ( x ) of Legendre type orthogonal with respect to constant weights w ∗ s = ( N + ) − on the grid x s of zeros of the polynomial P N + ( x ) .Note that earlier similar property was established with respect to classical orthogonal poly-nomials [5], [17]. Indeed, consider, e.g. the Hermite polynomials H n ( x ) [8]. They satisfy therelation H ′ n + ( x ) = ( n + ) H n ( x ) , n = , , , . . . . (1.16)Hence for any N the pair H N + ( x ) , H N ( x ) is the Sturmian pair. In turn, it follows that for any N the mirror-dual polynomials H ∗ n ( x ) are orthogonal with constant weights ( N + ) − on the grid x , x , . . . , x N of the roots of the Hermite polynomial H N + ( x ) [5], [17].In this case however, the grid x s of the roots of the Hermite polynomial H N + ( x ) is not ”clas-sical”: there is no explicit expression for these roots as a function of s . Instead, we considerthe following problem. Let us start with prescribed ”classical” grid x s with distinct real entries.This grid defines uniquely the monic polynomial P N + ( x ) . Consider then the monic polynomial P N ( x ) = ( N + ) − P ′ N + ( x ) of degree N . Clearly, the roots of polynomials P N ( x ) and P N + ( x ) areinterlacing. Together, the polynomials P N + ( x ) , P N ( x ) constitute the Sturmian pair. Our mainproblem will be deriving explicit expressions for the Jacobi matrices J (or J ∗ ) and for correspondingorthogonal polynomials P n ( x ) .By ”classical grid” we mean the orthogonality grids for classical orthogonal polynomials ofdiscrete variable [8]. The most general grid of such type should satisfy the linear difference equation[10], [14], [16] x s + + x s − − Ω x s = ν (1.17)with some real constants Ω and ν . The type of the grid depends on the parameter Ω. More exactly:(ia) if ∣ Ω ∣ > x s = C q s + C q − s + C , (1.18)where Ω = q + q − with some real q . In particular, when C = x s = C q − s + C . (1.19)(ib) the case ∣ Ω ∣ < q = exp ( iθ ) (see[12] for details).(ii) if Ω = x s = C s + C s + C . (1.20)In particular, when C = x s = C s + C . (1.21)(iii) if Ω = − x s = ( − ) s ( C s + C ) + C . (1.22)We restrict ourselves only with linear, exponential and quadratic grids and with a special typeof trigonometric grid. Other types of grids will be considered elsewhere.The paper is organized as follows. In Section 2, we consider generic properties of polynomials P n ( x ) , P ∗ n ( x ) corresponding to the Sturm sequence. In particular, we reinterpret Sylvester’s ap-proach to the Sturm algorithm in terms of orthogonal polynomials. In section 3, the linear grid x s = s is considered. In this case the mirror duals with respect to the Chebyshev-Legendre poly-nomials appear as general solution. In Section 4, we study the quadratic grid x s = s ( s + τ ) , where τ is the only essential parameter which characterizes the grid. For the two cases: τ = τ = x s = q − s is analyzed leading to the general solution related to the q-Hahn polynomials. In Section 6, two ALEXEI ZHEDANOV types of trigonometric grids are considered. They correspond to solutions in terms of Chebyshevpolynomials of the first and second kind. In concluding remarks in Section 7 we discuss some openproblems. 2.
Generic properties of Sturmian sequences
Before considering concrete examples, let us mention some generic properties of the Jacobimatrices
J, J ∗ and corresponding polynomials P n ( x ) , P ∗ n ( x ) . First of all, the affine transformationof the grid x s → αx s + β leads to the polynomials P n ( x ) with corresponding affine transformedargument x . For the Jacobi matrices J, J ∗ this transformation leads to shifting of the diagonalentries b n → b n + const and to rescaling the off-diagonal entries u n . Hence we can use this affinetransformation in order to reduce the grid to the most convenient form.Second, let us consider the moments c ∗ n corresponding to the dual weights w ∗ s c ∗ n = N ∑ s = w ∗ s x ns = ( N + ) − N ∑ s = x ns . (2.1)This formula indicates that the moments c ∗ n are nothing else than sum of powers of the grid points.For the classical grids these sums can be explicitly calculated. For example, for the linear grid(after its appropriate affine transformation) x s = s, s = , , , . . . , N we have that the moments c ∗ n coincide with Faullhaber sums c ∗ n = N ∑ s = s n . (2.2)These sums can be expressed in terms of the Bernoulli numbers (see e.g. [1]).We can relate the above results with Sylvester’s approach to the Sturm problem [13], [11].Indeed, let P n ( x ) be a finite system of orthogonal polynomials defined by the recurrence relation(1.3). One can introduce then the system of orthogonal polynomials of second kind R n ( x ) definedby [4] R n + ( x ) + b n + R n ( x ) + u n + ( x ) = xR n ( x ) , n = , , . . . , N, R = , R ( x ) = x − b . (2.3)Then one can construct the continued fraction [4] F ( z ) = R N ( z ) P N + ( z ) = z − b − u z − b − u z − b − ⋯ − u N z − b N . (2.4)The rational function F ( z ) plays the role of the Stieltjes transform of the orthogonality measure.Indeed, expanding F ( z ) in term of partial fractions, we have [4] F ( z ) = N ∑ s = w s z − x s (2.5)so that the weights w s are residues of F ( z ) at simple poles z = x s .On the other hand, by the Euclidean algorithm, we have the continued fractions F ∗ ( z ) = P N ( z ) P N + ( z ) = z − b N − u N z − b N − − u N − z − b N − − ⋯ − u z − b . (2.6)It is clear that the functions F ( z ) and F ∗ ( z ) are mirror conjugate. In more details, the function F ( z ) corresponds to the Jacobi matrix J while the function F ∗ ( z ) corresponds to the Jacobi matrix J ∗ . By definition, the simple poles x s are the same for both functions F ( z ) and F ∗ ( z ) . Hence, ifwe need only information about poles x s , we can use any of the Stieltjes functions F ( z ) or F ∗ ( z ) .But for the Sturmian pair P N ( x ) = ( N + ) − P ′ N + ( x ) , the function F ∗ ( z ) has extremely simpleexpansion in terms of partial fraction: F ∗ ( z ) = P ′ N + ( z )( N + ) P N + ( z ) = N + N ∑ s = ( z − x s ) . (2.7)In other words, formula (2.7) is equivalent to the statement that the ”mirror” weights w ∗ s are allequal one to another (1.15). And this was the key idea of Sylvester’s approach - to use the mirror-reflected continued fraction (2.6) instead of (2.4) [13]. We thus see that Sylvester’s approach hassimple and transparent interpretation in terms of orthogonal polynomials. LASSICAL STURMIAN SEQUENCES 5
As a direct application, let us consider the Hankel determinants constructed from abstractmoments c n ∆ n = det ⎛⎜⎜⎜⎝ c c . . . c n − c n c c . . . c n c n + . . . . . . . . . . . . . . .c n c n + . . . c n − c n ⎞⎟⎟⎟⎠ , ∆ ( ) n = ⎛⎜⎜⎜⎝ c c . . . c n c n + c c . . . c n + c n + . . . . . . . . . . . . . . .c n + c n + . . . c n c n + ⎞⎟⎟⎟⎠ . (2.8)The classical result from the theory of orthogonal polynomials is the Stieltjes criterion [4]: theconditions ∆ n > , ∆ ( ) n > n = , , , . . . (2.9)are equivalent to existence of a positive orthogonality measure on the semi-axis 0 < x < ∞ . Thismeans that there exists a positive measure dµ ( x ) such that c n = ∫ ∞ x n dµ ( x ) , n = , , , . . . (2.10)In our case this criterion means that all roots of the polynomial P N + ( x ) are positive. The abovecriterion remains true if one replaces the moments c n with the ”mirror” moments c ∗ n defined by(2.1). This is one of the main Sylvester’s results [11].Finally, note that if the grid x s is symmetric around zero, i.e. if x N − s = − x s , s = , , . . . N, (2.11)then the matrices J, J ∗ have zero diagonal entries, i.e. b n = , n = , , . . . , N . Equivalently, thismeans that the polynomials P n ( x ) and P ∗ n ( x ) are symmetric: P n ( − x ) = ( − ) n P n ( x ) . (2.12)This property can easily be derived from the observation that for symmetric grid x s (2.11) thecharacteristic polynomial P N + ( x ) is either odd or even depending on parity of N . The polynomial P N ( x ) has obviously opposite parity. Then by induction one can prove that all further polynomials P n ( x ) satisfy the property (2.12). Equivalently, this means that the Jacobi matrices J and J ∗ havezero diagonal entries [4]. 3. The linear grid
We start with the most simple case when the grid is linear. By an affine transformation it isalways possible to reduce the linear grid to the standard form x s = , , , . . . , N. (3.1)Then we arrive at the problem of finding orthogonal polynomials having constant weights w ∗ s = ( N + ) − on the linear grid (3.1). This problem was solved by Chebyshev [10]. It appears thatthese Chebyshev polynomials of Legendre type are special case of the Hahn polynomials (Notenevertheless, that the generic Hahn polynomials were introduced by Chebushev too [8]).In order to describe these Chebyshev-Legendre polynomials, recall first that generic monic Hahnpolynomial H n ( x ; α, β, N ) depend on 2 parameters α, β and on the integer parameter N . Explicitly,these polynomials are expressed in terms of hypergeometric function H n ( x ; α, β, N ) = κ n F ( − n, − x, n + α + β + − N, α + ) , (3.2)where κ n is the normalization coefficient κ n = ( − N ) n ( α + ) n ( n + α + β + ) n and where ( x ) n = x ( x + ) . . . ( x + n − ) is the Pochhammer symbol.The Hahn polynomials satisfy the three-term recurrence relation H n + ( x ) + b n H n ( x ) + u n H n − ( x ) = xH n ( x ) (3.3)with b n = A n + C n , u n = A n − C n , (3.4)where A n = ( n + α + β + )( n + α + )( N − n )( n + α + β + )( n + α + β + ) , C n = n ( n + α + β + N + )( n + β )( n + α + β + )( n + α + β ) . (3.5) ALEXEI ZHEDANOV
These polynomials are orthogonal on the linear lattice x s = s N ∑ s = W s H n ( s ; α, β, N ) H m ( s ; α, β, N ) = h n δ nm (3.6)with the weights W s = ν ( − N ) s ( α + ) s s ! ( − N − β ) s , (3.7)where ν = ( β + ) N ( α + β + ) N (3.8)is the normalization coefficient needed to fulfill the standard condition N ∑ s = W s = . (3.9)It is now easy to see that if α = β = W s = N ! ( N + ) ! = N + x s = s .Hence we have for the mirror-dual polynomials P ∗ n ( x ) the recurrence coefficients u ∗ n = n (( N + ) − n ) ( n − ) , b ∗ n = N / , n = , , . . . , N (3.11)The corresponding Sturmian polynomials P n ( x ) also belong to the class of Hahn polynomials.Indeed, it is easy to check that the transformation of the parameters˜ α = − N − − β, ˜ β = − N − − α (3.12)is equivalent to the mirror transform u n → u ∗ n = u N + − n , b n → b ∗ n = b N − n of the recurrence coeffi-cients of the Hahn polynomials. Hence H ∗ n ( x ; α, β, N ) = H n ( x, ˜ α, ˜ β, N ) (3.13)For the case of Chebyshev polynomials of Legendre type we have˜ α = ˜ β = − N − Proposition 2.
The Sturmian sequence of the polynomials P n ( x ) on the linear grid x s = s = , , . . . , N coincides with the set of the Hahn polynomials P n ( x ) = H n ( x, α, β, N ) , (3.15) where α = β = − N − . It is interesting to note that polynomials (3.15) are orthogonal on the grid x s = s with respectto the squared binomial distribution w s = ν ( N ! s ! ( N − s ) ! ) , (3.16)where ν = ( N ! ) ( N ) ! (3.17)Moreover, these polynomials satisfy the simple difference equation on the linear grid ( x − N ) ( P n ( x + ) − P n ( x )) + x ( P n ( x − ) − P n ( x )) = n ( n − N − ) P n ( x ) . (3.18) LASSICAL STURMIAN SEQUENCES 7 Quadratic grid
Consider the quadratic grid x s = a s + a s + a , where all parameters a , a , a are real. By anappropriate affine transformation it is always possible to reduce this grid to the form x s = s ( s + τ ) (4.1)with the only parameter τ which defines the type of the grid. Note that for the linear grid thereare no free parameters: all linear grids are affine equivalent to the standard one x s = s . For thequadratic grid we should distinguish different types of grids depending on the essential parameter τ . As in the previous section, we should first to construct the mirror-dual polynomials P ∗ m ( x ) ofLegendre type on the grid (4.1). This means that these polynomials should satisfy the orthogonalityrelation N ∑ s = P ∗ n ( s ( s + τ )) P ∗ m ( s ( s + τ )) = ( N + ) h ∗ n δ nm (4.2)It is natural to search the polynomials P ∗ n ( x ) among ”classical” polynomials orthogonal on thequadratic grid (4.1). Such polynomials are well known. The most general family of these polyno-mials - Racah polynomials - contains 4 parameters α, β, γ, δ [8]. One of these parameters, say α ,should be chosen as α = − N − P n ( x ; β, γ, δ, N ) = κ n F ( − n, − n + β − N, − s, s + γ + δ + − N, β + δ + , γ + ) , (4.4)where x ( s ) = s ( s + γ + δ + ) (4.5)The recurrence relation is P n + ( x ) + b n P n ( x ) + u n P n − ( x ) = xP n ( x ) , (4.6)with b n = − A n − C n , u n = A n − C n , (4.7)where A n = ( n + β − N )( n + β + δ + )( n + γ + )( n − N )( n + β − N )( n + β − N + ) ,C n = n ( n + β )( n + β − γ − N − )( n − δ − N − )( n + β − N )( n + β − N − ) (4.8)The orthogonality of the Racah polynomials can be presented as [8] N ∑ s = W s P n ( x ( s )) P m ( x ( s )) = h n δ nm , (4.9)where the weights are W s = M − ( − N ) s ( β + δ + ) s ( γ + ) s ( γ + δ + ) s (( γ + δ + )/ ) s s ! ( γ + δ + + N ) s ( − β + γ + ) s ( δ + ) s (( γ + δ + )/ ) s , (4.10)where the ”total mass” is M = ( − β ) N ( γ + δ + ) N ( + γ − β ) N ( δ + ) N . (4.11)The mirror-dual Rach polynomials P ∗ n ( x ) again belong to the Racah class and have the parameters β ∗ = − β, γ ∗ = δ, δ ∗ = γ. (4.12)Necessary condition to get all the weights W s not depending on s is to equate all parametersin the numerator Pochhammer symbols in (4.10) to corresponding parameters in the denominatorPochhammer symbols. There are several solutions. However, up to the reflection s → N − s andshifting the linear grid, there are basically only two solutions:(i) γ = − / , δ = / , β = N + /
2. In this case x s = s ( s + ) .(ii) γ = δ = − / , β = N + /
2. In this case x s = s . ALEXEI ZHEDANOV
The solution (ii) should be discarded because it leads to a singularity in the the expression (4.10).This singularity may be avoided if one takes appropriate limiting procedure δ = − / + ε, g = − / + ε with ε →
0. This procedure yields the weights (up to a normalization factor) W = , W s = , s = , , . . . N (4.13)which does not correspond to the Legendre case.We thus arrived at the only possible type of the quadratic grid x s = s ( s + ) .The recurrence coefficients are u ∗ n = n ( n − ) (( N + ) − n )) ( N + − n )( N + + n )( n + )( n − )( n − ) (4.14)and b ∗ n = ( N + / )( N + / ) ( n − − n − ) + ( N − n )( n + N + ) + N +
532 (4.15)Jacobi matrix J corresponding to the Sturm sequence on the same quadratic grid x s = s ( s + ) areobtained from the above coefficients by the mirror symmetry b n → b N − n , u n → u N + − n .We thus have the Proposition 3.
The Sturm sequence on quadratic grid x s = s ( s + ) coincides with the sequenceof the Racah polynomials R n ( x ; α, β, γ, δ ) with the parameters α = − N − , γ = / , δ = − / , β =− N − / . Consider now the quadratic grid with τ =
2, i.e. x s = s ( s + ) . If one puts δ = γ = / , β = N + / W s = M − ( x s + ) , (4.17)where M = N ( N + )( N + ) . (4.18)Let P ( ) ( x ) be the orthogonal polynomials corresponding to the Legendre-type measure W ( ) s = ( N + ) − on the grid x s = s ( s + ) . Then formula (4.17) indicates that the Racah polynomials P n ( x ) with δ = γ = / , β = N + / P ( ) ( x ) by the Christoffel transform [19] P n ( x ) = P ( ) n + ( x ) − V n P ( ) n ( x ) x + , V n = P ( ) n + ( − ) P ( ) n ( − ) . (4.19)Indeed, the Christoffel transform [19] is transformation of the orthogonal polynomials P n ( x ) → ˜ P n ( x ) which corresponds to the transformation of the orthogonality measure dµ ( x ) d ˜ µ ( x ) = ( x − a ) dµ ( x ) (4.20)with some real parameter a located outside the orthogonality interval.Equivalently, the Christoffel transform can be presented as˜ P n ( x ) = P n + ( x ) − V n P n ( x ) x − a , V n = P n + ( a ) P n ( a ) . (4.21)The transformed polynomials ˜ P n ( x ) satisfy the recurrence relation˜ P n + ( x ) + ˜ b n ˜ P n ( x ) + ˜ u n ˜ P n − ( x ) = x ˜ P n ( x ) (4.22)with the transformed recurrence coefficients [19]˜ u n = u n V n / V n − , ˜ b n = b n + + V n + − V n . (4.23)Conversely, the polynomials P n ( x ) can be obtained from the q -Hahn polynomials Q n ( x ) by theGeronimus transform [19] P n ( x ) = ˜ P n ( x ) − U n ˜ P n − ( x ) , (4.24)where U n = ϕ n ϕ n − . (4.25) LASSICAL STURMIAN SEQUENCES 9
The sequence φ n ( z ) is an arbitrary solution of the recurrence relation ϕ n + + ˜ b n ϕ n + ˜ u n ϕ n − = aϕ n . (4.26)We thus see that the polynomials of Legendre type on the quadratic grid x s = s ( s + ) can beobtained from the Racah polynomials ˜ P n ( x ) on the same grid by the Geronimus transform (4.24),where the parameters of the Racah polynomials are α = − N − , δ = γ = / , β = N + / x s = s ( s + ) we can use formulas(1.14) and (4.12). From these formula it is seen that the Sturmian polynomials are obtained fromthe Racah polynomials with the parameters α = N − , β = − N − / , γ = δ = / a = − Proposition 4.
The Sturm sequence P n ( x ) , n = , , . . . , N of the polynomials on the quadraticgrid x s = s ( s + ) coincides with polynomials obtained by the Christoffel transforms of the Racahpolynomials R n ( x ) ≡ P n ( x ; α, β, δ, γ ) with parameters α = N − , β = − N − / , γ = δ = / : P n ( x ) = R n + ( x ) − R n + (− ) R n (− ) R n ( x ) x + . (4.27)These polynomials do not belong to the Askey scheme [8] and hence they do not satisfy anysecond-order difference equation on the grid x s = s ( s + ) .5. Exponential grid
The exponential grid can be reduced by affine transformations to the canonical form x s = q − s , s = , , . . . N. (5.1)We assume that 0 < q < Q n ( x ; α, β, N ) = φ ( q − n , αβq n + , xq − N , αq ∣ q ; q ) . (5.2)They are orthogonal on the exponential grid (5.1) N ∑ s = W s Q n ( q − s ) Q m ( q − s ) = h n δ nm (5.3)with W s = M − ( αq ; q ) s ( q − N ; q ) s ( q ; q ) s ( β − q − N ; q ) s ( αβq ) − s (5.4)and M = ( αβq ; q ) N ( βq ; q ) N ( αq ) N , (5.5)where ( x ; q ) s stands for the q-Pochhammer symbol ( x ; q ) s = ( − x )( − xq ) . . . ( − xq s − ) . The q-Hahn polynomials satisfy the three-term recurrence relation Q n + ( x ) + b n Q n Q n ( x ) + u n Q n − ( x ) = xQ n ( x ) (5.6)with [8] u n = A n − C n , b n = − A n − C n , (5.7)where A n = ( − q n − N ) ( − aq n + ) ( − abq n + )( − abq n + ) ( − abq n + ) (5.8)and C n = − aq n − N ( − q n ) ( − bq n ) ( − abq n + N + )( − abq n + ) ( − abq n ) . (5.9)In contrast to the case of the ordinary Hahn polynomials, it is impossible to obtain the Legendre-type weights W s = ( N + ) − for q-Hahn polynomials . Nevertheless, we can achieve the desired result by putting α = β =
1. Then the weights become W s = M − q − s . (5.10)This means that the measure (5.10) is obtained from the uniform Legendre-type measure by mul-tiplying to the argument x which takes the values x ( s ) = q − s on the grid x ( s ) .Let P ( ) ( x ) be the orthogonal polynomials corresponding to the Legendre-type measure W ( ) s = ( N + ) − on the grid q − s . Then formula (5.10) indicates that the q-Hahn polynomials Q n ( x ) with α = β = P ( ) ( x ) by the Christoffel transform [19] Q n ( x ) = P ( ) n + ( x ) − V n P ( ) n ( x ) x , V n = P ( ) n + ( ) P ( ) n ( ) . (5.11)Conversely, the polynomials P ( ) n ( x ) can be obtained from the q -Hahn polynomials Q n ( x ) by theUvarov transform [15], which can be considered as a special case of the Geronimus transform [19] P ( ) n ( x ) = Q n ( x ) − U n Q n − ( x ) , (5.12)where U n = F n ( ) F n − ( ) . (5.13)The functions F n ( z ) are defined as F n ( z ) = ∫ Q n ( x ) z − x dµ ( x ) . (5.14)These functions are known to yield the second linear independent solution of the same recurrencerelation as for the orthogonal polynomials P n ( x ) [10] F n + ( z ) + b n F n ( z ) + u n F n − ( z ) = zF n ( z ) . (5.15)In our concrete case these functions can be calculated as F n ( ) = N ∑ s = W s Q n ( q − s ) − q − s = κ N ∑ s = Q n ( q − s ) (5.16)with some normalization constant κ which does not depend on n .We thus have the Proposition 5.
The Legendre type polynomials P ( ) n ( x ) on the exponential grid x s = q − s are givenby the Uvarov transform (5.12)-(5.13) of the q-Hahn polynomials Q n ( x ; α, β, N ) with α = β = ,where F n ( ) has expression (5.16). In order to obtain explicit expression for the Sturm polynomials on the exponential grid we noticethat the reflection Q n ( x ) → Q ∗ n ( x ) for q-Hahn polynomials is equivalent to the transformation a ∗ = b − q − N − , b ∗ = a − q − N − . (5.17)Moreover, from the formula (1.14) we see that the Uvarov transform of the polynomials P ∗ n ( x ) corresponds to the Christoffel transform of the polynomials P n ( x ) . Omitting obvious technicaldetails, we arrive at Proposition 6.
The Sturm sequence P n ( x ) , n = , , . . . , N of the polynomials on the exponentialgrid x s = q − s coincides with polynomials obtained by the Christoffel transforms of the q-Hahnpolynomials with parameters a = b = q − N − : P n ( x ) = Q n + ( x ) − Q n + ( ) Q n ( ) Q n ( x ) x , (5.18) where Q n ( x ) ≡ Q n ( x ; q − N − , q − N − , N ) . As a direct consequence of this proposition, using (4.23), we derive explicit expressions for therecurrence coefficients of the Sturm polynomials: u n = u ( ) n V n / V n − , b n = b ( ) n + + V n + − V n , (5.19)where V n = Q n + ( ) Q n ( ) (5.20) LASSICAL STURMIAN SEQUENCES 11 and where u ( ) n and b ( ) n are recurrence coefficients (5.7) of the q-Hahn polynomials with the pa-rameters a = b = q − N − .Note that in contrast to the case of linear grid, the polynomials (5.18) do not belong to theAskey scheme. Hence they don’t satisfy any second order difference eigenvalue equation.6. Trigonometric grids
In this section we consider two special cases of the trigonometric grids connected with roots ofunity.The first grid is given by x s = − cos ( ω ( s + / )) , s = , , , . . . , N, (6.1)where ω = πN + . (6.2)This trigonometric grid is related to the Chebyshev polynomials polynomials of the first kind T n ( x ) [9]. Indeed, these (monic) polynomials are defined as T = , T n ( x ) = − n cos ( θn ) , n = , , , . . . (6.3)with x = cos θ (6.4)They satisfy the recurrence relation T n + ( x ) + u n T n − ( x ) = xT n ( x ) , n = , , , . . . (6.5)where the recurrence coefficients are u = / , u n = / , n = , , . . . (6.6)The monic Chebyshev polynomials of the second kind are defined as U n ( x ) = − n sin ( θ ( n + )) sin θ , n = , , , . . . (6.7)They satisfy the same recurrence relation U n + ( x ) + u n U n − ( x ) = xU n ( x ) , n = , , , . . . (6.8)with u n = / n = , , . . . . Thus the only difference between recurrence relations forpolynomials T n ( x ) and U n ( x ) is expression of the first recurrence coefficient: u = / T n ( x ) and u = / U n ( x ) .There is obvious relation between these polynomials [9]: T ′ n + ( x ) = ( n + ) U n ( x ) , n = , , , . . . (6.9)The roots x s of the Chebyshev polynomial T N + ( x ) coincide with (6.1). On these roots theChebyshev polynomials T n ( x ) satisfy the finite orthogonality relation [9] N ∑ s = T n ( x s ) T m ( x s ) = ( N + ) h n δ nm . (6.10)Hence the Chebyshev polynomials of the first kind T n ( x ) satisfy orthogonality relation of Legendretype. This allows one to construct corresponding Sturmian orthogonal polynomials.Indeed, let us start with prescribed grid (6.1). Then the initial Sturmian polynomial P N + ( x ) coincides with the Chebyshev polynomial having zeros on this grid: P N + ( x ) = T N + ( x ) . Us-ing formula (6.9), we conclude that the companion Sturmian polynomial P N ( x ) coincides withthe Chebushev polynomial of the second kind: P N ( x ) = ( N + ) − P ′ N + ( x ) = U N ( x ) . Applyingthe Sturm algorithm, we then reconstruct the whole chain of Sturmian orthogonal polynomials P N − ( x ) , P N − ( x ) , . . . , P ( x ) .In order to recognize these polynomials P n ( x ) , we can use Proposition . The polynomials P n ( x ) are mirror-dual with respect to the polynomials T , T ( x ) , T ( x ) , . . . T N ( x ) . Hence we havethe recurrence coefficients for them b n = , n = , , , . . . , N, u = u = ⋅ ⋅ ⋅ = u N − = / , u N = / . (6.11)Comparing recurrence coefficients (6.11) we arrive at Proposition 7.
For the trigonometric grid (6.1) the two initial Sturmian polynomials are P N + ( x ) = T N + ( x ) and P N ( x ) = U N ( x ) . All further orthogonal polynomials coincide with the Chebyshevpolynomials of the second kind: P n ( x ) = U n ( x ) , n = , , . . . , N − . These polynomials satisfy therecurrence relation P n + ( x ) + u n P n − ( x ) = xP n ( x ) , n = , , . . . , N (6.12) with the recurrence coefficients u n given by (6.11). It is easy to find the orthogonality relation for the above polynomials P n ( x ) N ∑ s = P n ( x s ) P m ( x s ) w s = h n δ nm . (6.13)Indeed, we already know that the mirror-dual orthogonality relation is of Legendre type (6.10).Hence, by (1.14) we have w s = N + ( sin θ s sin ( θ s ( N + )) ) = N + θ s , (6.14)where θ s = π ( s + / ) N + . (6.15)These weights are normalized, i.e. N ∑ s = w s = . (6.16)As expected, the weights (6.14) are well known discrete orthogonality weights for the Chebyshevpolynomials U n ( x ) on the roots of the Chebyshev polynomial T N + ( x ) [9].The second case of the trigonometric grid coincides with zeros of the Chebyshev polynomial U N + ( x ) of the second kind: x s = − cos ( π ( s + ) N + ) , s = , , . . . , N. (6.17)In order to describe the corresponding Sturmian sequence of polynomials P n ( x ) , x = , , . . . , N we recall the well known properties of the ultrasperical polynomials. The monic ultrasphericalpolynomials C ( λ ) n ( x ) depend on one parameter λ and are defined as [8] C ( λ ) ( x ) = κ n F ( − n, n + λλ + / ∣ − x ) , (6.18)where the normalization coefficient is κ n = − n ( λ ) n ( λ ) n . (6.19)They satisfy the 3-term recurrence relation [8] C ( λ ) n + ( x ) + u n C ( λ ) n − ( x ) = xC ( λ ) n ( x ) (6.20)with u n = n ( n + λ − ) ( n + λ )( n + λ − ) . (6.21)The Chebyshev polynomials are special cases of the ultrapsherical polynomials T n ( x ) = C ( ) n ( x ) , U n ( x ) = C ( ) n ( x ) . (6.22)Moreover, there is simple formula [8] dC ( λ ) n ( x ) dx = nC ( λ + ) n − ( x ) (6.23)from which it follows that dU n + ( x ) dx = ( n + ) C ( ) n ( x ) . (6.24)We can now take derivatives from all members of the Sturmian sequence U , U ( x ) , . . . , U N ( x ) , T N + ( x ) corresponding to the previous type of trigonometric grid. We then have the sequence C ( ) ( x ) , C ( ) ( x ) , C ( ) ( x ) , . . . , C ( ) N ( x ) , U N + ( x ) , LASSICAL STURMIAN SEQUENCES 13 where all polynomials apart from the final one are the ultrapsherical polynomials C ( ) n ( x ) and thefinal polynomial P N + ( x ) coincides with the Chebyshev polynomial U N + ( x ) . The roots of thepolynomial P N + ( x ) are described by (6.17). Moreover, due to (6.24) we have ( N + ) P N ( x ) = P ′ N + ( x ) . Hence we have the needed Sturm sequence of the polynomials P n ( x ) satisfying therecurrence relation P n + ( x ) + u n P n − ( x ) = xP n ( x ) (6.25)with the recurrence coefficients u n = n ( n + ) ( n + )( n + ) , n = , , . . . , N − , u N = N ( N + ) . (6.26)We thus see that both trigonometric grids (6.1) and (6.17) correspond to elementary solutions ofthe Sturm sequence related with the Chebyshev polynomials T n ( x ) and U n ( x ) .7. Conclusion
We have derived explicit systems of finite orthogonal polynomials corresponding to classicalSturm problem for the pair of polynomials P ( x ) and P ′ ( x ) on prescribed grids: linear, exponentialand for special types of quadratic and trigonometric grids. In all the above cases, the correspondingpolynomials are related to classical polynomials from the Askey scheme [8].The open problems are constructing of Sturmian polynomials on other types of grids: Askey-Wilson and Bannai-Ito. Another interesting problem is related to the quadratic and trigonometricgrid. We were able to construct explicitly the Sturm polynomials for the grid x s = s ( s + ) and x s = s ( s + ) . What about generic quadratic grid x s = s ( s + τ ) with an arbitrary parameter τ ? Andwhat are other examples of trigonometric grids admitting explicit solution for the Sturm sequence?Another open problem arises in connection with Sturmian polynomials on the quadratic x s = s ( s + ) and exponential x s = q − s grids. As we already mentioned, these polynomials are obtainedfrom the classical ones (i.e. from the Racah and q-Hahn polynomials) by the Christoffel transform.This means that they are not ”classical” in sense of classification of the Aseky scheme [8]. Inparticular, they don’t satisfy a linear difference equation of second order. On the other hand, itis known that some Christoffel transformed classical polynomials of discrete variable do satisfydifference equations of higher orders [6]. One can ask whether or not the Sturmian polynomials onthe above lattices satisfy some difference equations of order 4 or more. Acknowledgments
The work is supported by the National Science Foundation of China(Grant No.11771015). The author thanks to Luc Vinet for discussion.
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