Stanley, R. P., "Differentiably finite power series." Eur. J. Comb. 1(1980), 175--188.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....1. All three terms on the right are regular at t = t 0 , hence so is f(x(t) Therefore f(x) should be regular at x = x 0 . This contradiction shows that f(x) is not a rational power series. 2 We conjecture that f(x) and therefore F (x; y) is not algebraic, and, moreover, not even D finite [9, 5]. Note that by replacing the initial conditions (16) by a i;0 = 2 ; a i;1 = 2 i 1 (i 0) a 0;k = 2 ; a 1;k = 2 (k 0) the solution of (15) changes to a i;k = 2 with rational generating function F (x; y) 1 Gamma2x) 1 Gamma2y) Acknowledgement. The author wishes to express ....

R. P. Stanley, Differentiably finite power series, European J. Combin. 1 (1980) 175--188.

.... = 0 (with at least a j such that P i;j 6= 0 for each i) An equivalent definition states that the space spanned by all the derivatives of F is finite dimensional over K (x) D finite functions have nice closure properties and are related to a lot of combinatorial problems (see Stanley s article [12] and Lipshitz s article [7] Definition 2 (Apex) The apex of H is the componentwise maximum of H [ f0g. For example, for the chess knight problem, one has H = f( Gamma2; 1) 1; Gamma2)g so the apex is (1; 1) and the starting point is (2; 2) If H = f( Gamma2; Gamma1) Gamma1; 2)g, then ....

Stanley (R. P.). -- Differentiably finite power series. European Journal of Combinatorics, vol. 1, n 2, 1980, pp. 175--188.

....d (x 1 ; x d )Z d (x 1 ; x d ) which leads to the result. This factorization generalizes the standard argument for counting 2D staircase SAP mentioned in the introduction. We will also consider the series Z (t) Z (t; t) which is also D finite (in one variable) [31]. For instance, Z 2 (t) is, according to (3) equal to (1 Gamma 4t) but it can alternatively be characterized by the initial condition Z 2 (0) 1 and the differential equation (1 Gamma 4t) 2 (t) Gamma 2 Similarly, the series Z 3 (t) is characterized by Z 3 (0) 1, Z ....

R. P. Stanley, Differentiably finite power series, Europ. J. Combinatorics 1 (1980) 175--188.

....formal power series, that is, power series F (x 1 ; x n ) m 1 ; mn f(m 1 ; m n )x m 1 1 Delta Delta Delta x mn n algebraic over the field K(x 1 ; x n ) of rational functions in the variables x i . Recent interest in algebraic generating functions [7] [11] has been connected with the fact that there are efficient algorithms for computing the coefficients f(m 1 ; m n ) Additionally, the minimal polynomial of F contains asymptotic information about these coefficients. The object of this paper is to prove: Theorem 1. Let K be a field with a ....

R. P. Stanley, Differentiably finite power series. European J. Combinatorics 1 (1980), 175--188. 8

....1. All three terms on the right are regular at t = t 0 , hence so is f(x(t) Therefore f(x) should be regular at x = x 0 . This contradiction shows that f(x) is not a rational power series. 2 We conjecture that f(x) and therefore F (x; y) is not algebraic, and, moreover, not even D finite [6, 2]. Note that by replacing the initial conditions (18) by a i;0 = 2 ; a i;1 = 2 (i 0) a 1;k = 2 (k 0) the solution of (17) changes to a i;k = 2 (1 Gamma2x) 1 Gamma2y) Acknowledgement. The author wishes to express his thanks to Philippe Flajolet, Bruno Salvy and Ivan Vidav ....

R. P. Stanley, Differentiably finite power series, European J. Combin. 1 (1980) 175--188.

....k F x k i Delta Delta Delta P 1 (x) F x i P 0 (x)F = 0 holds, where the polynomials P have their coefficients in A. The series F is transcendental if it is not algebraic. The coefficients of a D finite series are said to be P recursive. For properties of D finite series, see [22] for univariate series and [15] for the multivariate case. We shall clarify below the connection between our linearly recurrent sequences and P recursiveness. 8 4.1 From the recurrence relation to a functional equation Let us now transform our recurrence relation into a functional equation ....

R. P. Stanley, Differentiably finite power series, European J. Combin. 1 (1980) 175--188.

....functions come in quite different disguises. It turns out that sum and product of two holonomic functions are again holonomic, and the corresponding holonomic (differential or recurrence) equations can be constructed from the given holonomic equations by linear algebra (see [1] 2] [24], 23] As an example, we consider both the sum and the product of the functions f(x) arcsin x and g(x) e x . Here are their holonomic equations: DE1: SimpleDE(arcsin(x) x,F) DE1 : x 1) x 1) # 2 #x 2 F(x) # #x F(x) x =0 DE2: SimpleDE(exp(x) x,F) DE2 : # #x ....

R. P. STANLEY, Differentiably finite power series, European J. Combin. 1 (1980), pp. 175--188.

....is P recursive. The formula (3) shows that Catalan numbers form a P recursive sequence. We shall see that fa 5 (n)g n0 and fa 6 (n)g n0 are P recursive as well. On the other hand we prove now that fa 1 (n)g n0 and fa 3 (n)g n0 are not P recursive. The basic reference for P recursivity is Stanley [7]. There it is proved that the P recursivity of fa(n)g n0 is equivalent to the fact that F (x) P n0 a(n)x n satisfies, for some m 1 polynomials R 0 (x) Rm (x) 2 C[x] Rm (x) 6j 0, the differential equation Rm (x)F (m) x) R 1 (x)F 0 (x) R 0 (x)F (x) 0: 12) This ....

R. P. Stanley, Differentiably finite power series, Europ. J. Combinatorics 1 (1980), 175--188.

....algorithm requires only rational operations (such as gcd and resultant computations) but no factorization. 1 Introduction Let F be a field of characteristic zero. We will denote by F the ring of all sequences over F , with addition and multiplication defined term wise. Following Stanley [6] we identify two sequences if they agree from some point on. Formally, we are working in the quotient ring F=J where J is the ideal of sequences with only finitely many non zero terms. Hence, all equalities of the form lhs(n) rhs(n) are to be interpreted as valid for all large enough n. ....

R.P. Stanley, Differentiably finite power series, European J. Combin. 1 (1980) 175 -- 188. 8

....1 ; z 2 ; z r ) A sequence f n1;n2 ; n r is holonomic iff its generating function f(z 1 ; z 2 ; z r ) X n1;n2 ; n r fn1;n2 ; n r z n1 1 z n2 2 Delta Delta Delta z nr r is holonomic. The major closure theorem here is due to Stanley, Lipschitz, and Zeilberger [4, 5, 7, 8]. Theorem 6 (Holonomic Closure) Holonomic functions are closed under sums, Cauchy products, Hadamard products, diagonals, algebraic substitutions, integration, differentiation, direct and inverse Laplace transforms. Theorem 7 (Holonomic Asymptotics) A holonomic sequence f n is asymptotic to a ....

Stanley (R. P.). -- Differentiably finite power series. European Journal of Combinatorics, vol. 1, 1980, pp. 175--188.

....by W n;m (x) Gamma( Gamma2m) Gamma(1=2 Gamma m Gamma n) M n;m (x) Gamma(2m) Gamma(1=2 m Gamma n) M n; Gammam (x) 1] 13.1. 32) W n;m (x) is represented as sum of products, the recurrence equation can be obtained from (4) and the recurrence equation of the Gamma function (see e.g. [25], 32] 17] 24] 2 We note that similarly, one can obtain holonomic recurrence equations for the Hahn type polynomials h ( n (x; N) see [22] x2.4) and p n (x; fi; fl; ffi) see [7] x 10.23) Note further that some of the above recurrence equations have appeared in the literature. ....

....into the form F 0 n (x) 1 x i (n Gamma x) F n (x) Gamma (n Gamma 1) F n Gamma1 (x) j (17) using (16) This is a derivative rule of the form (13) Therefore the functions F n (x) form an admissible family of order two. 4 Properties of Admissible Families It is well known (see e.g. [25], 32] 17] 24] that if the functions f n ; g n satisfy holonomic recurrence equations of order m and l, respectively, then the sum and product satisfy holonomic recurrence equations of order m l, and m l, respectively. We call the two functions f n and g n sum independent ....

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Stanley, R. P.: Differentiably finite power series. Europ. J. Combinatorics 1, 1980, 175-- 188.

....radius of F is the smallest root of its denominator. Its value is easily determined numerically and found to be about 0.360102. 2 4. 2 Holonomy In the transcendental case, one can also discuss the holonomic character of the generating function F (z) A series is said to be holonomic, or D finite [25], if it satisfies a linear differential equation with polynomial coefficients in z. Equivalently, its coefficients f n satisfy a linear recurrence relation with polynomial coefficients in n. Consequently, given a sequence f n , the ordinary generating function P n f n z n is holonomic if and ....

R. P. Stanley. Differentiably finite power series. European Journal of Combinatorics, 1:175--188, 1980.

.... = 0 (with at least a j such that P i;j 6= 0 for each i) 3 An equivalent definition states that the space spanned by all the derivatives of F is finite dimensional over K (x) D finite functions have nice closure properties and are related to a lot of combinatorial problems (see Stanley s article [12] and Lipshitz s article [7] Definition 2 (Apex) The apex of H is the componentwise maximum of H [ f0g. For example, for the chess knight problem, one has H = f( Gamma2; 1) 1; Gamma2)g so the apex is (1; 1) and the starting point is (2; 2) If H = f( Gamma2; Gamma1) Gamma1; 2)g, then ....

Stanley (R. P.). -- Differentiably finite power series. European Journal of Combinatorics, vol. 1, n 2, 1980, pp. 175--188.

....p 0 (n) p 1 (n) p k (n) such that k X i=0 p i (n)a(n i) 0 for all nonnegative integers n. The fundamental fact relating these two concepts is that a(n) is Precursive if and only if its generating function P 1 n=0 a(n)x n is D finite. We refer the reader to STANLEY [20] for the proof of this and other basic facts. In this paper we show that counting sequences for certain combinatorial problems which can be expressed as coefficients of symmetric functions are P recursive. To do this we need a multivariable generalization of the theory of D finiteness and ....

....: ym , as long as it is well defined as a formal power series. iv) If P (x) is a polynomial in x 1 ; x 2 ; xn then e P (x) is D finite. The proofs of these statements are straightforward, and are similar to 6 SYMMETRIC FUNCTIONS proofs for the one variable case given by STANLEY [20]. See also LIPSHITZ [10] We need one further fact about D finite power series in several variables, due to LIPSHITZ [9] which is somewhat harder to prove. If A(x) P a(i 1 ; i n )x i 1 1 Delta Delta Delta x i n n and B(x) P b(i 1 ; i n )x i 1 1 Delta Delta ....

STANLEY (R.P.). --- Differentiably finite power series, European J. Combin., t. 1, 1980, p. 175--188.

....transcendental functions come in quite different disguises. It turns out that sum and product of two holonomic functions are again holonomic, and the corresponding holonomic (differential or recurrence) equations can be constructed from the given holonomic equations by linear algebra ( 1] 2] [22], 21] As an example, we consider both the sum and the product of the functions f(x) arcsin x and g(x) e x . Here are their holonomic equations: DE1: SimpleDE(arcsin(x) x,F) DE1 : x Gamma 1) x 1) 2 x 2 F(x) x F(x) x = 0 DE2: SimpleDE(exp(x) x,F) DE2 : ....

Stanley, R. P.: Differentiably finite power series. Europ. J. Combinatorics 1, 1980, 175--188.

....and discrete orthogonal polynomials. 1 Holonomic Functions A homogeneous linear differential equation m X k=0 p k (x) f (k) x) 0 with polynomial coefficients p k (x) is called holonomic, as is the corresponding f(x) Holonomic functions have nice algebraic properties: In 1980 Stanley [22] proved by algebraic arguments that the sum and product of holonomic functions, and their composition with algebraic functions, in particular with rational functions and rational powers, form holonomic functions, again. By iterative differentiation and the use of Gaussian elimination, one can ....

....the second computation is obvious. The automatic computation of holonomic recurrence equations for expressions is supported through the procedure HolonomicRE[expr,a[k] Examples are In[21] HolonomicRE[ n k 2) k,a[n] 2 2 Out[21] 1 n) a[n] 1 3 n n ) a[1 n] n a[2 n] 0 In[22]: HolonomicRE[ n k 2) k,a[k] 3 Out[22] k (1 k) 3 k) a[k] 2 2 (1 k) 1 3 k k ) 3 3 k k ) a[1 k] 2 k (2 k) a[2 k] 0 The work with special functions for which holonomic recurrence equations exist, is possible. In[23] HolonomicRE[LegendreP[k,x] P[k] Out[23] ....

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Stanley, R. P.: Differentiably finite power series. Europ. J. Combinatorics 1, 1993, 175-- 188.

....takes values 4 and 7. Let s 2 (n) be the sum of the binary digits of the integer n. Then, the formal power series P 2 s 2 (n) ff n n X n is transcendental over Q(X) hint: reduce modulo 3, and apply Cobham s theorem to the Sigma1 Thue Morse sequence) 6 Binomial series In 1980 Stanley [55] conjectured that the series P n0 Gamma 2n n Delta t X n , where t is a positive integer, is algebraic only for t = 1 (for which its value is 1= p 1 Gamma 4X ) And he proved, but only for even t s, that this series is indeed transcendental. The conjecture was proved by Flajolet in ....

R. P. Stanley, Differentiably finite power series, Eur. J. Comb. 1 (1980) 175--188.

.... = 0 (with at least a j such that P i;j 6= 0 for each i) 3 An equivalent definition states that the space spanned by all the derivatives of F is finite dimensional over K (x) D finite functions have nice closure properties and are related to a lot of combinatorial problems (see Stanley s article [11]) Definition 2 (Apex) The apex of H is the componentwise maximum of H [ f0g. For example, for the chess knight problem, one has H = f( Gamma2; 1) 1; Gamma2)g so the apex is (1; 1) and the starting point is (2; 2) If H = f( Gamma2; Gamma1) Gamma1; 2)g, then the apex is (0,2) Theorem ....

Stanley (R. P.). -- Differentiably finite power series. European Journal of Combinatorics, vol. 1, n 2, 1980, pp. 175--188.

....as being valid for all but finitely many n (in short: for almost all n) Define on S(IF) by requiring that (a J) Ea J for all a 2 IF IN . Then is an automorphism of S(IF) and S(IF) is a difference extension ring of IF(x) The elements of S(IF) are called the germs of sequences over IF [15]. To simplify notation, we will identify the germ a J 2 S(IF) with its representative sequence a 2 IF IN . In this domain the q difference equation Ly = 0 where y = y n ) 1 n=0 2 S(IF) translates into ae X i=0 p i (q n )y n i = 0 for almost all n. In particular, the first order ....

....it follows that X k f(n; k) 1 2 X k (f(n; k) f(n; Gammak) 1 2 X k (1 q k )f(n; k) The extra factor 1 q k increases the chance that the q Zeilberger algorithm finds a recurrence of lower order. Indeed, now one gets by applying qZeil the minimal recurrence of order 1. In[15]: qZeil[ 1)k (1 qk) 2 q(k(3k 1) 2) qBinomial[2n,n k,q] k, Infinity,Infinity, n, 1] n 1 2 n Out[15] SUM[n] 1 q ) 1 q ) SUM[ 1 n] Paule s method is of special importance with respect to the theory of q WZ pairs [16] There are various applications, 10] or [14] where summing ....

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R. P. Stanley, Differentiably finite power series, European J. Combin. 1 (1980) 175--188. 21

....2 ; z r ) A sequence f n1 ;n2 ; n r is holonomic iff its generating function f(z 1 ; z 2 ; z r ) P n1 ;n2 ; n r f n1 ;n2 ; n r z n1 1 z n2 2 Delta Delta Delta z n r r is holonomic. The major closure theorem here is due to Stanley, Lipschitz, and Zeilberger [59, 60, 73, 83]. 14 Theorem 8 (Holonomic Closure) Holonomic functions are closed under sums, products, Hadamard products, diagonals, algebraic substitutions, integration, differentiation, direct and inverse Laplace transforms. Coefficients sequences enjoy the corresponding closure properties. For instance, ....

Stanley, R. P. Differentiably finite power series. European Journal of Combinatorics 1 (1980), 175--188.

....other means, namely by taking the derivative of the reversion of the series, whose generating function is 1 (1 2x) 1 x) To describe other possible extensions of our approach, we recall some definitions. A series y(x) with coefficients in K, is said to be differentiably finite or D finite [Stanley 1980] if it satisfies some nontrivial linear differential equation p 0 (x)y p 1 (x)y 0 Delta Delta Delta p k (x)y (k) 0 (4.1) with coefficients p j (x) 2 K[x] A series y = y(x) is said to be constructible differentially finite or CDF [Bergeron and Reutenauer 1990] if, for some k ....

R. P. Stanley, "Differentiably finite power series", Europ. J. Combin. 1 (1980), 175--188.

....being valid for all but finitely many n (in short: for almost all n) Define on S(IF) by requiring that (a J) Ea J for all a 2 IF IN . Then is an automorphism of S(IF) and S(IF) is a difference extension ring of IF(x) The elements of S(IF) are called the germs of sequences over IF [15]. To simplify notation, we will identify the germ a J 2 S(IF) with its representative sequence a 2 IF IN . 9 In this domain the q difference equation Ly = 0 where y = y n ) 1 n=0 2 S(IF) translates into ae X i=0 p i (q n )y n i = 0 for almost all n. In particular, the first order ....

....it follows that X k f(n; k) 1 2 X k (f(n; k) f(n; Gammak) 1 2 X k (1 q k )f(n; k) The extra factor 1 q k increases the chance that the q Zeilberger algorithm finds a recurrence of lower order. Indeed, now one gets by applying qZeil the minimal recurrence of order 1. 17 In[15]: qZeil[ 1)k (1 qk) 2 q(k(3k 1) 2) qBinomial[2n,n k,q] k, Infinity,Infinity, n, 1] n 1 2 n Out[15] SUM[n] 1 q ) 1 q ) SUM[ 1 n] Paule s method is of special importance with respect to the theory of q WZ pairs [16] There are various applications, 10] or [14] where summing the ....

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R. P. Stanley, Differentiably finite power series, European J. Combin. 1 (1980) 175--188.

....i (x) v i (x a) Therefore it suffices to consider existence of nice expansions around x = 0, then translate the results back to expansions around x = a. It is well known that for a fixed a, the sequence (wn ) 1 n=0 satisfies a linear recurrence equation with polynomial coefficients (cf. [8]) Denote x k = ae 1; k = 0; x(x Gamma 1) Delta Delta Delta (x Gamma k 1) k 0; x k = ae 1; k = 0; x(x 1) Delta Delta Delta (x k Gamma 1) k 0; and rewrite (4) as X i;j c ij x i z (j) x) 0 (5) where c ij = 0 unless 0 i d and 0 j r, not all c d;j are ....

R. P. Stanley (1980): Differentiably finite power series, European J. Combin. 1, 175--188. 7

.... S d (x 1 ; x d )Z d (x 1 ; x d ) which leads to the result. This factorization generalizes the standard argument for counting 2D staircase SAP mentioned in the introduction. We will also consider the series Z (t) Z (t; t) which is also D finite (in one variable) [31]. For instance, Z 2 (t) is, according to (3) equal to (1 Gamma 4t) Gamma1=2 , but it can alternatively be characterized by the initial condition Z 2 (0) 1 and the differential equation (1 Gamma 4t) Z 0 2 (t) Gamma 2 Z 2 (t) 0: Similarly, the series Z 3 (t) is characterized by ....

R. P. Stanley, Differentiably finite power series, Europ. J. Combinatorics 1 (1980) 175--188.

....following properties: it is an algebra and in particular is closed under sum and product; it contains all algebraic functions and is closed under algebraic substitution; it is closed under Hadamard (i.e. term wise) product and under diagonal. All these results are proved by Stanley in [22]. P recursive sequences also form an algebra and possess corresponding properties. These interesting properties have led Salvy and Zimmermann to implement the Gfun Maple package described in [21] This package manipulates sequences, linear recurrence equations or linear differential equations and ....

....cases of holonomic functions, D finite functions in Section 1.1 and P recursive sequences in Section 1.2. We recall proofs of their closure properties and of the D FINITE FUNCTIONS, P RECURSIVE SEQUENCES AND HOLONOMIC SYSTEMS 5 fundamental equivalence theorem; detailed proofs can be found in [22, 17, 18]. Then, in Section 1.3, we extend the definitions to holonomic systems, that involve both D finiteness and P recursiveness. Throughout Sections 1.1 and 1.2, K is a field of characteristic zero. This field K will usually be Q, R or C in practice; it may also be a finitely generated extension of Q ....

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Stanley, R. P. Differentiably finite power series. European Journal of Combinatorics 1 (1980), 175--188.

....SC 95 13 (May 1995) The Identification Problem for Transcendental Functions Wolfram Koepf koepf zib berlin.de Abstract In this article algorithmic methods are presented that have essentially been introduced into computer algebra within the last decade. The main ideas are due to Stanley [34] and Zeilberger [40] 43] Some of them had already been discovered in the last century (see e.g. 4] 5] but because of the complexity of the underlying algorithms have fallen into oblivion. The combination of these ideas leads to a solution of the identification problem for a large class of ....

....is homogeneous, linear, of order one, and has polynomial coefficients. We can generalize this observation [40] A continuous function of one variable f(x) is holonomic, if it satisfies a homogeneous linear differential equation with polynomial coefficients. By linear algebra arguments, Stanley [34] showed that sums and products of holonomic functions and the composition with algebraic functions also form holonomic functions. It is remarkable that exactly 100 years ago, Beke [4] 5] could describe algorithms to generate holonomic differential equations for the sum and product of f and g ....

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Stanley, R. P.: Differentiably finite power series. Europ. J. Combinatorics 1, 1980, 175-- 188.

....Bessel functions (see e.g. 1] Ch. 9 11) all kinds of orthogonal polynomials (see e.g. 16] 17] and functions of hypergeometric type (see [6] Not all elementary transcendental functions, however, are simple, the simplest example of which probably is the tangent function tan x, compare [15], Example 2.5. Theorem 1 The functions tan x and sec x do not satisfy simple differential equations. Proof: It is easily seen that the tangent function f(x) tan x satisfies the nonlinear differential equation f 0 = 1 f 2 : 1) Differentiation of (1) yields after further substitution of ....

....and therefore has dimension n q. 2 Lemma 1 shows immediately that for h : f ffi x 1=q the linear space L h over K[x] generated by h; h 0 ; h 00 ; is a subset of L f (declared in Lemma 2) and so by Lemma 2 is of dimension n q. Thus we have proved the following Theorem (compare [15], Theorem 2.7, 14] Maple function algebraicsubs) Theorem 2 Let f be simple of order n, and let q 2 IN. Then f ffi x 1=q is simple of order n q. 2 Whereas Theorem 2 gives a complete answer with regard to the existence of a simple differential equation, in [15] Theorem 2.7, it is shown that ....

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Stanley, R. P.: Differentiably finite power series. Europ. J. Combinatorics 1, 1993, 175--188.

....If there exist polynomials p 0 (x) p 1 (x) Delta Delta Delta p d (x) so that p d 6= 0 and p d (x)u (d) x) p d Gamma1 (x)u (d Gamma1) x) Delta Delta Delta p 1 (x)u 0 (x) p 0 (x)u(x) 0; 2) then we say that u is d finite. Here u (j) d j u dx j ) It is well known [13] that a function f(n) is P recursive if and only if its ordinary generating function u(x) F (x) P n0 f(n)x n is d finite. Another, smaller class of formal power series is that of algebraic series. We say that the series v(x) 2 C [ x] is algebraic if there exist polynomials p 0 (n) p 1 ....

R. P. Stanley, Differentiably Finite Power Series, European Journal of Combinatorics, 1 (1980), 175-188.

.... satisfies a linear recurrence with polynomial coefficients (a holonomic recurrence) It can be shown that the generating function (either ordinary or exponential) of a holonomic sequence is holonomic and that reciprocally the sequence of Taylor coefficients of a holonomic function is holonomic [17]. This correspondence is implemented in the procedures diffeqtorec and rectodiffeq: diffeqtorec This procedure translates a holonomic equation c k (z)y (k) z) Delta Delta Delta c 1 (z)y 0 (z) c 0 (z)y(z) b(z) 0 for the function y(z) into a holonomic recurrence for the ....

.... 2 2 d 2 3 d (z z ) y(z) z z ) y(z) 1 z) y(z) 1 dz 2 dz 2.1 Building up equations The class of holonomic functions and sequences enjoys nice closure properties. In particular we have the following theorem. Theorem 1 (Closure) [6, 13, 17, 22] (a) algebraic functions are holonomic; b) the sum of two holonomic functions is holonomic; c) the Cauchy product of two holonomic functions is holonomic (convolution of the sequences) d) the Hadamard product of two holonomic functions is holonomic (term wise product of the sequences) e) if ....

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Stanley, R. P. Differentiably finite power series. European Journal of Combinatorics 1 (1980), 175--188.

....capability of Maple s dsolve differential equation solver was used on that occasion. In this context, an interesting class of functions is that of combinatorial holonomic systems of Zeilberger [88] which, in the univariate case, reduces to the class of D finite function described by Stanley [77]. In other words, we could also regard as known the solution Y (z) of any equation d X j=0 Q j (z) d j dz j Y (z) 0; where the Q j are rational functions, and proper initial conditions completely determine Y (z) This class has rich closure properties. To a large extent, the ....

Stanley, R. P. Differentiably finite power series. European Journal of Combinatorics 1 (1980), 175--188.

....coefficients. We can generalize this observation [42] A continuous function of one variable f(x) is holonomic, if it satisfies a homogeneous linear differential equation with polynomial coefficients; we call such a differential equation also holonomic. By linear algebra arguments, Stanley [36] showed that sums and products of holonomic functions and the composition with algebraic functions also form holonomic functions. This can be seen as follows: Assume f and g satisfy holonomic differential equations of order n and m, respectively. We consider the linear space L f of functions with ....

....the classical families of orthogonal polynomials 1 and many other special functions form holonomic functions [1] These depend on several variables, and we will discuss this situation in x 4. On the other hand, there are functions that are not holonomic, like the tangent function tan x (s. [36], 25] The identification problem for expressions involving nonholonomic functions can only be treated after preprocessing the input. If, for example, we want to verify the addition formula for the tangent function tan (x y) tan x tan y 1 Gamma tan x tan y by the given method, then we ....

Stanley, R. P.: Differentiably finite power series. Europ. J. Combinatorics 1, 175--188 (1980).

....by the cancellation of the denominator: it is finite and nonzero; its value is easily determined numerically and found to be about 0.360102. 2 In the transcendental case, one can also discuss the holonomic character of the generating function F (z) A series is said to be holonomic, or D finite [17], if it satisfies a linear differential equation with polynomial coefficients in z. Holonomic functions include algebraic functions, and have a finite number of singularities. Example 9 is holonomic, while Example 10 is not, as it has infinitely many singularities. Amongst the simplest systems ....

R. P. Stanley. Differentiably finite power series. Electronic Journal of Combinatorics, 1:175--188, 1980.

....interest to computer algebra and combinatorics, since they can be specified by a finite amount of information: the coefficients and a finite number of initial conditions. This has led D. Zeilberger to generalize the notions of P recursive sequences and D finite functions studied by R. Stanley [24] into a notion of P finiteness [33] In several variables, a function is P finite when the vector space generated by its derivatives has finite dimension over the field of rational functions. Similarly, a sequence is P finite when the vector space generated by its shifts has finite dimension ....

....univariate and multivariate cases. In particular, algebraic functions are holonomic and an algorithm to compute differential equations from the polynomial equation exists [8] Also, the composition of a holonomic function with algebraic functions is again holonomic and equations can be computed [17, 24]. Holonomic functions are defined as solutions of differential equations with polynomial (or equivalently, rational) coefficients. There is in fact no enlargement of the class if we allow algebraic functions as coefficients: a function that satisfies a rectangular system with algebraic ....

Stanley, R. P. Differentiably finite power series. European Journal of Combinatorics 1 (1980), 175--188.

....coefficients. The basic theory and algebraic background are given in Sections 1.2 and 1.3, respectively, where we also present some famous members of the holonomic family. The results discussed there and in Section 1. 4 are well known, a comprehensive exposition of the subject was given by Stanley [Sta80]. In fact the algebraic properties of holonomic functions and sequences, presented in Section 1.4 are the main motivation to work in the holonomic universe: Most elementary operations like addition, termwise or Cauchy) multiplication and several univariate transformations preserve holonomicity. ....

.... p d (x)f (d) x) 0: 1.2.1) CHAPTER 1. THE HOLONOMIC UNIVERSE 8 The nonnegative integer d denotes the order of the holonomic equation. The degree of equation (1.2.1) is given by maxfdeg(p i (x) j0 i dg. In literature holonomic functions are also called differentiably finite, D finite [Sta80], Lip89] GKP94] or simple functions [Koe92] Subsequently, we will sometimes write f and p instead of f(x) and p(x) if the indeterminate x is clear from the context. If we extend our working domain from the ring K[ x] to K( x) the field of formal Laurent series, i.e, series of the form ....

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R. P. Stanley. Differentiably finite power series. European Journal of Combinatorics, 1:175--188, 1980. BIBLIOGRAPHY 87

....is sorted, and, in this case, exhibits one of its pre images. Concerning the enumeration of sorted permutations, we shall give a functional equation (of a strange sort) satisfied by their generating function. So far, we have not been able to say whether this generating function is D finite [18], or at least differentiably algebraic [9] ffl Sorted and (one stack) sortable permutations We can describe these permutations by any of the three equivalent conditions: 2 Pi(S n ) and Pi( 12 : n, is the image by Pi of a two stack sortable permutation, is an inner ....

R. P. Stanley, Differentiably finite power series, Europ. J. Combinatorics, 1 (1980) 175--188.

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R. Stanley, Differentiably finite power series, European J. Combinatorics 1 (1980), 175--188.

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Stanley, R. P., "Differentiably finite power series." Eur. J. Comb. 1(1980), 175--188.

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R. P. Stanley. Differentiably finite power series. Electronic Journal of Combinatorics, 1:175--188, 1980.

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R.P. Stanley, Differentiably finite power series, Europ. J. Combinatorics 1 (1980), 175--188.

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