Strong q-log-convexity of the Eulerian polynomials of Coxeter groups
aa r X i v : . [ m a t h . C O ] S e p Strong q -log-convexity of the Eulerian polynomials ofCoxeter groups ∗ Lily Li Liu , Bao-Xuan Zhu School of Mathematical Sciences, Qufu Normal University, Qufu 273165, PR China School of Mathematical Sciences, Jiangsu Normal University, Xuzhou 221116, PR China
Abstract
In this paper we prove the strong q -log-convexity of the Eulerian polynomials ofCoxeter groups using their exponential generating functions. Our proof is based onthe theory of exponential Riordan arrays and a criterion for determining the strong q -log-convexity of polynomials sequences, whose generating functions can be givenby the continued fraction. As consequences, we get the strong q -log-convexity of theEulerian polynomials of types A n , B n , their q -analogous and the generalized Eulerianpolynomials associated to the arithmetic progression { a, a + d, a + 2 d, a + 3 d, . . . } ina unified manner. MSC:
Keywords:
Eulerian polynomials; Coxeter groups; Strong q -log-convexity; Riordanarray; Continued fraction The Eulerian polynomials P ( W, q ), which enumerate the number of descents of a(finite) Coxeter group W , is one of the classical polynomials in combinatorics. Duringtheir long history, they arised often in combinatorics and were extensively studied (see [5,6, 7, 13] and references therein). In recent years, there has been a considerable amount ofinteresting extensions and modifications devoted to these polynomials (see [2, 3, 11, 18,19, 20, 22, 25] for instance). In fact, Brenti showed that it is enough to study the Eulerianpolynomials for irreducible Coxeter groups [4, 5]. For Coxeter groups of type A n , it isknown that these polynomials coincide with the classical Eulerian polynomials, whoseproperties have been well studied from a combinatorial point of view [13, 16, 18, 19, 22].Some properties of the classical Eulerian polynomials can be generalized to the Eulerian ∗ This work was supported in part by the National Natural Science Foundation of China (Nos.11201260, 11201191), the Key Project of Chinese Ministry of Education (No. 212098), the Special-ized Research Fund for the Doctoral Program of Higher Education of China (No. 20113705120002), andthe National Science Foundation of Shandong Province of China (No. ZR2011AL018).
Email addresses: [email protected] (L.L. Liu), [email protected] (B.-X. Zhu) B n , such as recurrence relations, the reality of zeros, generatingfunctions, unimodality and total positivity properties [2, 5, 12, 20]. In this paper, usingtheir exponential generating functions, we present the strong q -log-convexity of manyEulerian polynomials of Coxeter groups, which on one hand also generalizes the strong q -log-convexity of the classical Eulerian polynomials [25], on the other hand will give thestrong q -log-convexity of types A n , B n , their q -analogues and the generalized Eulerianpolynomials associated to the arithmetic progression { a, a + d, a + 2 d, a + 3 d, . . . } in aunified manner.Let q be an indeterminate. For two real polynomials f ( q ) and g ( q ), denote f ( q ) > q g ( q )if the difference f ( q ) − g ( q ) has only nonnegative coefficients as a polynomial of q . Wesay that a real polynomial sequence { f n ( q ) } n > is called q -log-convex if f n − ( q ) f n +1 ( q ) > q f n ( q )for n >
1, and it is strongly q -log-convex if f m − ( q ) f n +1 ( q ) > q f m ( q ) f n ( q )for all n > m >
1. Clearly, the strong q -log-convexity of polynomials sequences impliesthe q -log-convexity. However, the converse dose not follows.As we know that many famous polynomials sequences, such as the Bell polynomi-als [10, 19], the classical Eulerian polynomials [19, 25], the Narayana polynomials [9],the Narayana polynomials of type B [8] and the Jacobi-Stirling numbers [17, 26], are q -log-convex. Furthermore, almost all of these polynomials sequences are strongly q -log-convex [10, 17, 25]. In this paper we give the strong q -log-convexity of many Eulerianpolynomials. Our proof relies on the theory of exponential Riordan arrays and a criterionof Zhu [25] for determining the strong q -log-convexity of polynomials sequences, whosegenerating functions can be given by the continued fraction.This paper is organized as follows. In section 2, using the theory of exponential Riordanarrays and orthogonal polynomials, we first give the continued fraction of the ordinarygenerating function of the polynomials sequence, whose exponential generating functiongeneralizes the exponential generating function of many Eulerian polynomials. Then weobtain the strong q -log-convexity of the polynomials sequence using the continued fractionand a criterion of Zhu [25]. As applications, we obtain the strong q -log-convexity of theEulerian polynomials of Coxeter groups, including the Eulerian polynomials of types A n , B n , their q -analogues defined by Foata and Sch¨utzenberger [16] and Brenti [5] respectively,and the generalized Eulerian polynomials associated to the arithmetic progression { a, a + d, a + 2 d, a + 3 d, . . . } [24] in a unified manner in section 3. In section 4, we presentsome conjectures and open problems. Finally, in the Appendix, we can obtain a quickintroduction to the exponential Riordan arrays and the orthogonal polynomials used inthis paper. q -log-convexity In this section, we first give the continued fraction of the ordinary generating functionof the polynomials sequence { T n ( q ) } n > , whose exponential generating function general-izes the exponential generating functions of many Eulerian polynomials. Then using the2ontinued fraction and a criterion of Zhu [25], we prove the strong q -log-convexity of thepolynomials sequence { T n ( q ) } n > . Theorem 2.1.
Suppose that the exponential generating function of the polynomials se-quence { T n ( q ) } n > has the following simple expression g ( x ) = X n > T n ( q ) n ! x n = (cid:18) (1 − q ) e a (1 − q ) x − qe d (1 − q ) x (cid:19) b , (2.1) for a, b, d ∈ R . Then the ordinary generating function of { T n ( q ) } n > can be given by thecontinued fraction h ( x ) = X n > T n ( q ) x n = 11 − s ( q ) x − t ( q ) x − s ( q ) x − t ( q ) x − s ( q ) x − t ( q ) x − s ( q ) x − · · · , (2.2) where s i ( q ) = ( di + ab ) + ( di + bd − ab ) q and t i +1 ( q ) = d ( i + 1)( i + b ) q (2.3) for i > . In order to prove this theorem, we need three lemmas. Using the theory of the ex-ponential Riordan arrays, the first lemma presents that the production matrix P of theexponential Riordan array L = [ g ( x ) , f ( x )], where g ( x ) is the exponential generatingfunction of { T n ( q ) } n > given by (2.1), is tri-diagonal. Lemma 2.1.
The production matrix P of the exponential Riordan array L = [ g ( x ) , f ( x )] = "(cid:18) (1 − q ) e a (1 − q ) x − qe d (1 − q ) x (cid:19) b , e d (1 − q ) x − d [1 − qe d (1 − q ) x ] , for a, b, d ∈ R , is tri-diagonal.Proof. In order to get the production matrix P , it suffices to calculate r ( x ) and c ( x ).Recall that r ( x ) = f ′ ( ¯ f ( x )) , c ( x ) = g ′ ( ¯ f ( x )) g ( ¯ f ( x )) , where ¯ f ( x ) is the compositional inverse of f ( x ).By the direct calculation, we have f ′ ( x ) = (1 − q ) e d (1 − q ) x [1 − qe d (1 − q ) x ] . Note that the compositional inverse of f ( x ) satisfies f ( ¯ f ( x )) = e d (1 − q ) ¯ f − d [1 − qe d (1 − q ) ¯ f ] = x. f ( x ) = 1 d (1 − q ) ln (cid:18) dx dqx (cid:19) . Hence r ( x ) = f ′ ( ¯ f ( x )) = (1 + dx )(1 + dqx ) = 1 + d (1 + q ) x + d qx . On the other hand, we have g ′ ( x ) = b (cid:18) (1 − q ) e a (1 − q ) x − qe d (1 − q ) x (cid:19) b − (1 − q ) e a (1 − q ) x [ a + ( d − a ) qe d (1 − q ) x ][1 − qe d (1 − q ) x ] . So c ( x ) = g ′ ( ¯ f ( x )) g ( ¯ f ( x )) = b (1 − q )[ a + ( d − a ) qe d (1 − q ) ¯ f ]1 − qe d (1 − q ) ¯ f = ab (1 + dqx ) + b ( d − a ) q (1 + dx )= b [ a + ( d − a ) q ] + bd qx. Thus the production matrix P of L is tri-diagonal, where P = s ( q ) 1 0 0 0 0 · · · t ( q ) s ( q ) 1 0 0 0 · · · t ( q ) s ( q ) 1 0 0 · · · t ( q ) s ( q ) 1 0 · · · t ( q ) s ( q ) 1 · · · t ( q ) s ( q ) · · · ... ... ... ... ... ... . . . , (2.4)with s i ( q ) and t i +1 ( q ) given by (2.3).The second lemma constructs a family of orthogonal polynomials related to the pro-duction matrix P of the exponential Riordan array L = [ g ( x ) , f ( x )]. Lemma 2.2.
Suppose that the production matrix P of an exponential Riordan array L is tri-diagonal as above (2.4). Then we can construct a family of orthogonal polynomials Q n ( x ) defined by Q n ( x ) = [ x − s n − ( q )] Q n − ( x ) − t n − ( q ) Q n − ( x ) , (2.5) with Q ( x ) = 1 and Q ( x ) = x − s ( q ) , where coefficients s n − ( q ) and t n − ( q ) are givenby the expression (2.3).Proof. In order to construct the family of orthogonal polynomials Q n ( x ), it suffices to getthe coefficient matrix A such that Q ( x ) Q ( x ) Q ( x ) Q ( x )... = A xx x ... . (2.6)4nd by the condition and the Favard’s Theorem 5.1 in Appendix, we will get that theorthogonal polynomials Q n ( x ) satisfies the following P Q ( x ) Q ( x ) Q ( x ) Q ( x )... = s ( q ) 1 0 0 0 0 · · · t ( q ) s ( q ) 1 0 0 0 · · · t ( q ) s ( q ) 1 0 0 · · · t ( q ) s ( q ) 1 0 · · · ... ... ... ... ... ... . . . Q ( x ) Q ( x ) Q ( x ) Q ( x )... = xQ ( x ) xQ ( x ) xQ ( x ) xQ ( x )... . (2.7)Then we have that the coefficient matrix A satisfies P A xx x ... = A xx x x ... = A ¯ I xx x ... , (2.8)where ¯ I = ( δ i +1 ,j ) i,j > . Since the polynomials sequence { x k } k > is linear independence.So the coefficient matrices of the first and last polynomials in (2.8) are equal, i.e., P A = A ¯ I . Since P = L − ¯ L, ¯ I = ¯ LL − . So we have that the coefficient matrix A will satisfy L − ¯ LA = A ¯ LL − . Thus we can obtain that L − is a coefficient matrix of the orthogonalpolynomials Q n ( x ). The proof of the lemma is complete. Remark . From the proof of Lemma 2.2, we have that the coefficient matrix of theorthogonal polynomials Q n ( t ) is L − = " dx ) abd (1 + dqx ) bd − abd , d (1 − q ) ln (cid:18) dx dqx (cid:19) , which has been shown by Barry [3]. However our proof is more natural and based on thealgebraic method.The last lemma, obtained by Barry [3], gave the connection between the productionmatrix and the moment sequence of orthogonal polynomials. Lemma 2.3 ([3]) . Let L , T n ( q ) and Q n ( x ) be as above. Then we have { T n ( q ) } n > is themoment sequence of the associated family of orthogonal polynomials Q n ( x ) . Now we can obtain that the ordinary generating function of { T n ( q ) } n > is given bythe continued fraction (2.2) from Theorem 5.2, which proves Theorem 2.1.Then we can present the strong q -log-convexity of { T n ( q ) } n > using the followingcriterion of Zhu [25]. Theorem 2.2 ([25, Proposition 3.13]) . Given two sequences { s i ( q ) } i > and { t i +1 ( q ) } i > of polynomials with nonnegative coefficients, let X n > D n ( q ) x n = 11 − s ( q ) x − t ( q ) x − s ( q ) x − t ( q ) x − s ( q ) x − t ( q ) x − s ( q ) x − · · · . f s i ( q ) s i +1 ( q ) > q t i +1 ( q ) for all i > , then the sequence { D n ( q ) } n > is strongly q -log-convex. The main result of this section is the following.
Theorem 2.3.
The polynomials sequence { T n ( q ) } n > defined by (2.1) forms a strongly q -log-convex sequence for b > and d > a > Proof.
By Theorem 2.1, if the exponential generating function of { T n ( q ) } n > has theexpression (2.1), then we have the ordinary generating function of { T n ( q ) } n > can begiven by the continued fraction (2.2). Note that s i ( q ) = ( di + ab ) + ( di + bd − ab ) q and t i +1 ( q ) = d ( i + 1)( i + b ) q for i > s i ( q ) s i +1 ( q ) − t i +1 ( q )= (( di + ab ) + ( di + bd − ab ) q )(( di + d + ab ) + ( di + d + bd − ab ) q ) − d ( i + 1)( i + b ) q = ( di + ab )( di + d + ab ) + ( di + bd − ab )( di + d + bd − ab ) q +(( di + ab )( di + d + bd − ab ) + ( di + bd − ab )( di + d + ab ) − d ( i + 1)( i + b )) q = ( di + ab )( di + d + ab ) + ( di + bd − ab )( di + d + bd − ab ) q +(( di + ab )( di + d + bd − ab ) + abd ( b − − a b ) q > q ( di + ab )( di + d + ab ) + ( di + bd − ab )( di + d + bd − ab ) q + ( ab d − a b ) q > q . The first and second inequalities hold by conditions i, b > d > a >
0. Hence thepolynomials sequence { T n ( q ) } n > forms a strongly q -log-convex sequence by Theorem 2.2 Given a finite Coxeter group W , define the Eulerian polynomials of W by P ( W, q ) = X π ∈ W q d W ( π ) , where d W ( π ) is the number of W -descents of π . We refer the reader to Bj¨orner [4] forrelevant definitions.For Coxeter groups of type A n , it is known that P ( A n , q ) = A n ( q ) /q , the shiftedEulerian polynomials, whose strong q -log-convexity was obtained by Zhu [25]. Since theexponential generating function of { A n ( q ) } n > and { P ( A n , q ) } n > is X n > A n ( q ) x n n ! = (1 − q )1 − qe x (1 − q ) (3.1)and X n > P ( A n , q ) x n n ! = (1 − q ) e x (1 − q ) − qe x (1 − q ) (3.2)6espectively (see [13, p. 244]). So from Theorem 2.1, we have X n > A n ( q ) x n = 11 − x − qx − (2 + q ) x − qx − (3 + 2 q ) x − qx − (4 + 3 q ) x − · · · , with s i ( q ) = i + ( i + 1) q and t i +1 ( q ) = ( i + 1) q for i > X n > P ( A n , q ) x n = 11 − qx − qx − (1 + 2 q ) x − qx − (2 + 3 q ) x − qx − (3 + 4 q ) x − · · · , with s i ( q ) = ( i + 1) + iq and t i +1 ( q ) = ( i + 1) q for i > Proposition 3.1.
The polynomials P ( A n , q ) and A n ( q ) form strongly q -log-convex se-quences respectively. In [16], Foata and Sch¨utzenberger introduced a q -analog of the classical Eulerian poly-nomials defined by A n ( q ; t ) := X π ∈ S n q exc ( π )+1 t c ( π ) , where exc ( π ) and c ( π ) denote the numbers of excedances and cycles in π respectively.It is clear that A n ( q ; 1) = A n ( q ) is precisely the classical Eulerian polynomial. Brentishowed that the exponential generating function of { A n ( q ; t ) } n > is given by X n > A n ( q ; t ) x n n ! = (cid:18) (1 − q ) e x (1 − q ) − qe x (1 − q ) (cid:19) t . (3.3)So from Theorem 2.1, we have X n > A n ( q ; t ) x n = 11 − tx − tqx − ( t + 1 + q ) x − t + 1) qx − ( t + 2 + 2 q ) x − t + 2) qx − ( t + 3 + 3 q ) x − · · · . Here s i ( q ) = ( t + i ) + iq and t i +1 ( q ) = ( i + 1)( t + i ) q for i > Proposition 3.2.
The polynomials A n ( q ; t ) form a strongly q -log-convex sequence for t > . B n , suppose that the Eulerian polynomials of type B n P ( B n , q ) = n X k =0 B n,k q k , where B n,k is the Eulerian numbers of type B n counting the elements of B n with k B -descents. Then the Eulerian numbers of type B n satisfy the recurrence B n,k = (2 k + 1) B n − ,k + (2 n − k + 1) B n − ,k − . (3.4)Hence the Eulerian polynomials of type B n satisfy the recurrence P ( B n , q ) = [(2 n − q + 1] P ( B n − , q ) + 2 q ( q − P ′ ( B n − , q ) . (3.5)It is well known that P ( B n , q ) have only real zeros (see [5, 20] for instance). Note that theexponential generating function of the Eulerian polynomials of type B n has the followingexpression X n > P ( B n , q ) x n n ! = (1 − q ) e x (1 − q ) − qe x (1 − q ) (3.6)(see [5, Theorem3.4] and [12, Corollary3.9]). Hence from Theorem 2.1, we have thegenerating function of the Eulerian polynomials of type B n is given by X n > P ( B n , q ) x n = 11 − (1 + q ) x − qx − q ) x − qx − q ) x − qx − q ) x − · · · . Here s i ( q ) = (2 i + 1)(1 + q ) and t i +1 ( q ) = 4( i + 1) q for i > . Thus the strong q -log-convexity of P ( B n , q ) follows from Theorem 2.3. Proposition 3.3.
The polynomials P ( B n , q ) form a strongly q -log-convex sequence. From the definitions, if a sequence of polynomials is strongly q -log-convex, then it is q -log-convex. So we have the following corollary immediately. Corollary 3.1.
The polynomials P ( B n , q ) form a q -log-convex sequence.Remark . From Liu and Wang [19, Theorem 4.1], we can also get Corollary 3.1 usingrecurrences (3.4) and (3.5).Brenti [5] defined a q -analogue of the polynomials P ( B n , q ) by B n ( q ; t ) := X π ∈ B n q d B ( π ) t N ( π ) , where N ( π ) := |{ i ∈ [ n ] , π ( i ) < }| . In particular, if t = 1, then B n ( q ; 1) = P ( B n , q ), theEulerian polynomials of type B n . And if t = 0, then B n ( q ; 0) = A n ( q ), the classical Eu-lerian polynomials. He showed that the exponential generating function of { B n ( q ; t ) } n > has the following expression X n > B n ( q ; t ) x n n ! = (1 − q ) e x (1 − q ) − qe x (1 − q )(1+ t ) . (3.7)8o from Theorem 2.1, the generating function of B n ( q ; t ) is given by X n > B n ( q ; t ) x n = 11 − s ( q ) x − t ( q ) x − s ( q ) x − t ( q ) x − s ( q ) x − t ( q ) x − s ( q ) x − · · · . Here s i ( q ) = ( t + 1) i + 1 + [( t + 1)( i + 1) − q and t i +1 ( q ) = [( t + 1)( i + 1)] q for i > . Thus the strong q -log-convexity of B n ( q ; t ) follows from Theorem 2.3. Proposition 3.4.
The polynomials B n ( q ; t ) form a strongly q -log-convex sequence for t > . Recently, Xiong, Tsao and Hall [24] defined the general Eulerian numbers A n,k ( a, d )associated with an arithmetic progression { a, a + d, a + 2 d, a + 3 d, . . . } as A n,k ( a, d ) = ( − a + ( k + 2) d ) A n − ,k ( a, d ) + ( a + ( n − k − d ) A n − ,k − ( a, d ) , where A , − = 1 and A n,k = 0 for k > n or k − . In particular, when a = d = 1, A n,k (1 ,
1) = A n,k , the classical Eulerian numbers which enumerating the number of A n with k − { a, a + d, a + 2 d, a + 3 d, . . . } can be defined as P n ( q, a, d ) = n − X k = − A n,k ( a, d ) q k +1 . It is shown that the exponential generating function of { P n ( q, a, d ) } n > has the followingexpression X n > P n ( q, a, d ) x n n ! = (1 − q ) e ax (1 − q ) − qe dx (1 − q ) . (3.8)So from Theorem 2.1, the generating function of { P n ( q, a, d ) } n > is given by X n > P n ( q, a, d ) x n = 11 − s ( q ) x − t ( q ) x − s ( q ) x − t ( q ) x − s ( q ) x − t ( q ) x − s ( q ) x − · · · , with s i ( q ) = ( di + a ) + ( di + d − a ) q and t i +1 ( q ) = ( d ( i + 1)) q for i > q -log-convexity of P n ( q, a, d ) follows from Theorem 2.3. Proposition 3.5.
The general Eulerian polynomials P n ( q, a, d ) associated with an arith-metic progression { a, a + d, a + 2 d, a + 3 d, . . . } form a strongly q -log-convex sequence for d > a > . Concluding remarks and open problems
Let a , a , a , . . . be a sequence of nonnegative numbers. The sequence is called log-convex (respectively log-concave ) if for k > a k a k − a k +1 (respectively a k > a k − a k +1 ).Let { a ( n, k ) } k n be a triangular array of nonnegative numbers. Define a linear trans-formation of sequences by z n = n X k =0 a ( n, k ) x k , n = 0 , , , . . . . (4.1)We say that the linear transformation (4.1) preserve log-convexity if it preserves the log-convexity of sequences, i.e., the log-convexity of { x n } implies that of { z n } . We also saythat corresponding triangle { a ( n, k ) } k n preserve log-convexity. Liu and Wang [19]obtained the binomial transformation, the Stirling transformations of the first and secondkind preserve log-convexity respectively. They also proposed the following conjecture,which is still open now. Conjecture 4.1 ([19]) . The Eulerian transformation z n = P nk =0 A n,k x k preserve log-convexity. Similarly, we can raise the following problem related to the Eulerian polynomials oftype B n . Conjecture 4.2.
Let z n = n X k =0 B n,k x k (4.2) denote the Eulerian transformation of type B n . Then the transformation (4.2) preserveslog-convexity. The exponential Riordan array [1, 14, 15] denoted by L = [ g ( x ) , f ( x )], is an infi-nite lower triangular matrix whose exponential generating function of the k th column is g ( x )( xf ( x )) k /k ! for k = 0 , , , . . . , where g (0) = 0 = f (0). An exponential Riordan array L = ( l i,j ) i,j > can also be characterized by two sequences { c n } n > and { r n } n > such that l , = 1 , l i +1 , = X j > j ! c j l i,j , l i +1 ,j = 1 j ! X k > j − k !( c k − j + jr k − j +1 ) l i,j , for i, j > { c n } n > and { r n } n > the c − and z − sequences of L respectively. Associated to each exponential Riordan array L = [ g ( x ) , f ( x )], there is amatrix P = ( p i,j ) i,j > , called the production matrix , whose bivariate generating functionis given by e xy [ c ( x ) + r ( x ) y ] , where c ( x ) = g ′ ( ¯ f ( x )) g ( ¯ f ( x )) := X n > c n x n , r ( x ) = f ′ ( ¯ f ( x )) := X n > r n x n . et al. [14] obtained the elements of production matrix P = ( p i,j ) i,j > satisfying p i,j = i ! j ! ( c i − j + jr i − j +1 ) . Assume that c − = 0 . Note that P = L − ¯ L, ¯ I = ¯ LL − , where ¯ L is obtained from L with the first row removed and ¯ I = ( δ i +1 ,j ) i,j > , where δ i,j isthe usual Kronecker symbol.The following well-known results establish the relationship among the orthogonal poly-nomials, three-term recurrences, recurrence coefficients and the continued fraction of thegenerating function of the moment sequence. The first result is the well-known ”Favard’sTheorem”. Theorem 5.1 ([21, Th´eor´eme 9 on p. I-4], or [23, Theorem 50.1]) . Let { p n ( x ) } n > be a sequence of monic polynomials with degree n = 0 , , , . . . respectively. Then thesequence { p n ( x ) } n > is (formally) orthogonal if and only if there exist sequences { α n } n > and { β n } n > with β n = 0 such that the three-term recurrence p n +1 ( x ) = ( x − α n ) p n ( x ) − β n p n − ( x ) holds, for n > , with initial conditions p ( x ) = 1 and p ( x ) = x − α . Theorem 5.2 ([21, Propersition 1 (7) on p. V-5], or [23, Theorem 51.1]) . Let { p n ( x ) } n > be a sequence of monic polynomials, which is orthogonal with respect to some linear func-tional L . For n > , let p n +1 ( x ) = ( x − α n ) p n ( x ) − β n p n − ( x ) , be the corresponding three-term recurrence which is guaranted by Favard’s theorem. Thenthe generating function h ( x ) = ∞ X k =0 µ k x k for the moments µ k = L ( x k ) satisfies h ( x ) = µ − α x − β x − α x − β x − α x − β x − α x − · · · . References [1] P. Barry,
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