Intoduction ON THE RATIONALITY OF THE In this note we give a - PowerPoint PPT Presentation

Intoduction ON THE RATIONALITY OF THE In this note we give a formula for the generating function of the solution of MULTIDIMENSIONAL RECURSIVE SERIES а multidimensional difference equation under the assumption that the generating function of the initial data is known. We also state the necessary and sufficient condition for rationality of the generating function. Alexander LYAPIN Siberian Federal University Richard Stanley in his book «Enumerative combinatorics» gives a hierarchy of «the most useful» classes of the generating functions (GF): 07/24/2012 D -finite ⊃ algebraic ⊃ rational. Alexander LYAPIN (SibFU) On the rationality of GF 07/24/2012 1 / 10 Alexander LYAPIN (SibFU) On the rationality of GF 07/24/2012 2 / 10 Let C = { α } , where α = ( α 1 , . . . , α n ) , be a finite subset of the positive octant Z n + of the integer lattice Z n , f : Z n + → C and let m = ( m 1 , m 2 , . . . , m n ) ∈ C . Moreover for all α ∈ C the condition De Moivre considered the recursive series as the power series F ( z ) = f (0) + f (1) z + . . . + f ( k ) z k + . . . with the constant coefficients ( ∗ ) α 1 � m 1 , . . . , α n � m n f (0) , f (1) , . . . that make recursive sequence { f ( n ) } , n = 0 , 1 , 2 , . . . satisfying the difference equation be fulfilled. The problem Cauchy c 0 f ( x + m ) + c 1 f ( x + m − 1) + . . . + c i f ( x + m − i ) + . . . + c m f ( x ) = 0 , The problem Cauchy is to find the solution f ( x ) of the difference equation with some constant coefficients c i ∈ C , where 0 ≤ i ≤ m . (we use a multidimensional notation) � c α f ( x + α ) = 0 , (1) In 1722 he proved that the power series F ( z ) are rational functions. α ∈ C which coincides with the some given function ϕ : X m → C on the set X m = Z n � m + Z n � + \ («initial data»). + Alexander LYAPIN (SibFU) On the rationality of GF 07/24/2012 3 / 10 Alexander LYAPIN (SibFU) On the rationality of GF 07/24/2012 4 / 10

It’s well known that this Cauchy problem has a unique solution. Let J = ( j 1 , ..., j n ) , where j k ∈ { 0 , 1 } , k = 1 , ..., n , is an ordered set of zeros and ones. With every such set J we associate the face Γ J of the M. Bousquet-M´ elou, M. Petkovˇ sek, Linear recurrences with constant n -dimensional integer parallelepiped coefficients: the multivariate case, DM, 225, 51-75. Π m = { x ∈ Z n : 0 � x k � m k , k = 1 , ..., n } E. Leinartas, Multiple Laurent Series and Difference Equations. Siberian Mathematical Journal, 2004, Volume 45, Number 2, 321-326. as follows: E. Leinartas, Multiple Laurent series and fundamental solutions of Γ J = { x ∈ Π m : x k = m k , if j k = 1 , and x k < m k , if j k = 0 } . linear difference equations, Siberian Mathematical Journal, Vol. 48, No. 2, pp. 268–272. Alexander LYAPIN (SibFU) On the rationality of GF 07/24/2012 5 / 10 Alexander LYAPIN (SibFU) On the rationality of GF 07/24/2012 6 / 10 Theorem GF of «inital data» ϕ ( x ) The generating function F ( z ) = � z x +1 of the solution of the difference The function x ∈ Z n ϕ ( x ) + � Φ( z ) = equation (1) is z x + I x ∈ X m � � � F ( z ) P ( z ) = Φ τ, J ( z ) P τ ( z ) , where P τ ( z ) = c α z α is the generation function of the initial data of the difference equation (1). J τ ∈ Γ J α � m α � τ GF of «inital data» can be represented as the sum c α z α is the characteristic polynomial of the difference and P ( z ) = � α ∈ C � Φ( z ) = Φ J ( z ) , equation (1). J where ϕ ( τ + Jy ) � � Φ J ( z ) = Φ τ, J ( z ) , Φ τ, J ( z ) = z τ + Jy + I . y � 0 τ ∈ Γ J Alexander LYAPIN (SibFU) On the rationality of GF 07/24/2012 7 / 10 Alexander LYAPIN (SibFU) On the rationality of GF 07/24/2012 8 / 10

Example Theorem ϕ ( x ) The generating function F ( z ) = � z x +1 of the solution of the difference Bloom’s srtings x ∈ Z n + Bloom studies the number of singles in all the 2 x x -length bit strings, equation (1) is where a single is any isolated 1 or 0, i.e., any run of length 1. Let r ( x , y ) be � � � F ( z ) P ( z ) = Φ τ, J ( z ) P τ ( z ) , where P τ ( z ) = c α z α the number of n -length bit strings beginning with 0 and having y singles. τ ∈ Γ J α � m J α � τ D.M.Bloom, Singles in a Sequence of Coin Tosses, The College c α z α is the characteristic polynomial of the difference and P ( z ) = � Mathematics Journal, 29(1998), 307-344. α ∈ C equation (1). Corollary The generating function F ( z ) of the solution of the difference equation (1) is rational if and only if the generating function Φ( z ) of the initial data is rational. Alexander LYAPIN (SibFU) On the rationality of GF 07/24/2012 8 / 10 Alexander LYAPIN (SibFU) On the rationality of GF 07/24/2012 9 / 10 Example Computation Bloom’s srtings Bloom studies the number of singles in all the 2 x x -length bit strings, Φ 0 , 0 = 1 P 0 , 0 = z 2 w − zw − z − w , zw , where a single is any isolated 1 or 0, i.e., any run of length 1. Let r ( x , y ) be P 1 , 0 = z 2 w − zw − w , the number of n -length bit strings beginning with 0 and having y singles. Φ 1 , 0 = 0 , P 0 , 1 = z 2 w − zw − z , Φ 0 , 1 = 0 , D.M.Bloom, Singles in a Sequence of Coin Tosses, The College 1 P 1 , 1 = z 2 w , Mathematics Journal, 29(1998), 307-344. Φ 1 , 1 = z 2 w 2 , 1 The Cauchy problem P 2 , 0 = z 2 w − zw − w , Φ 2 , 0 = zw ( z 2 − z − 1) , The sequence r ( x , y ) satisfies the difference equation r ( x + 2 , y + 1) − r ( x + 1 , y + 1) − r ( x + 1 , y ) − r ( x , y + 1) + r ( x , y ) = 0 . with the «initail data» It’s easy! ϕ (0 , 0) = 1 , ϕ (1 , 0) = 0 , ϕ ( x , 0) = ϕ ( x − 1 , 0) + ϕ ( x − 2 , 0) , x � 2 , z − 1 F ( z ) = ϕ (1 , 1) = 1 , ϕ (0 , y ) = 0 , y � 1 and ϕ (1 , y ) = 0 , y � 2 . z 2 w − zw − z − w + 1 Alexander LYAPIN (SibFU) On the rationality of GF 07/24/2012 9 / 10 Alexander LYAPIN (SibFU) On the rationality of GF 07/24/2012 10 / 10 J = (0 , 0) , Γ (0 , 0) = { x 1 < 2 , x 2 < 1 }