Asymptotics of the Coefficients of Bivariate Analytic Functions with Algebraic Singularities Torin Greenwood June 9, 2015 AofA’15 1 / 33
Overview � Goal: Starting with the closed form for a generating function F ( z ), approximate [ z r ] F ( z ) as r → ∞ . � The coefficients [ z r ] F ( z ) count something useful. 2 / 33
Overview � Goal: Starting with the closed form for a generating function F ( z ), approximate [ z r ] F ( z ) as r → ∞ . � The coefficients [ z r ] F ( z ) count something useful. � Cauchy Integral Formula & Contour Deformations 2 / 33
Overview � Goal: Starting with the closed form for a generating function F ( z ), approximate [ z r ] F ( z ) as r → ∞ . � The coefficients [ z r ] F ( z ) count something useful. � Cauchy Integral Formula & Contour Deformations � Look at F with algebraic singularities. � The branch cuts will cause problems! 2 / 33
Overview � Goal: Starting with the closed form for a generating function F ( z ), approximate [ z r ] F ( z ) as r → ∞ . � The coefficients [ z r ] F ( z ) count something useful. � Cauchy Integral Formula & Contour Deformations � Look at F with algebraic singularities. � The branch cuts will cause problems! � Multivariate! Use the method from Pemantle and Wilson’s book. � Can’t use residues here. 2 / 33
The Procedure in One Dimension � Begin with the Cauchy Integral Formula: 1 ˆ [ z n ] F ( z ) = F ( z ) z − n − 1 dz 2 π i C 3 / 33
The Procedure in One Dimension � Begin with the Cauchy Integral Formula: 1 ˆ [ z n ] F ( z ) = F ( z ) z − n − 1 dz 2 π i C � Expand C until it gets stuck on a singularity of F ( z ). Away from the singularity, expand beyond it. 3 / 33
The Procedure in One Dimension � Begin with the Cauchy Integral Formula: 1 ˆ [ z n ] F ( z ) = F ( z ) z − n − 1 dz 2 π i C � Expand C until it gets stuck on a singularity of F ( z ). Away from the singularity, expand beyond it. � The z − n term forces decay away from the singularity. So, analyze the integrand near the singularity. 3 / 33
220 PHILIPPE FLAJOLET AND ANDREW ODLYZKO totic expansion as n -- , Thus the binomial coefficients (2.1), as well as their main asymptotic equivalents in (2.2), form an asymptotic scale. There is in fact a general form of (2.2). PROPOSITION 1. The binomial coefficients expressing [zn]( z) have an asymp- a{0,1,2, "’’}, (2.3) [Znl(1--Z)a’" 1+ Ia(--a kl where (2.4) e )= 2k (-1)lXk, t(a+ 1)(a+ 2)-" .(c+l) l=k with k,l_ 0 Proposition 1, although it would probably follow by close inspection of Stirling’s formula, is most easily proved by techniques introduced in 3, so that we delay the proof until then. We also observe, incidentally, that in (2.1)-(2.3 a may be complex: If c + it, we have cos (t log n) sin (t log n) ]. [Znl(1--Z) I’( -r it) In that case, the main term in (2.2), (2.3) is of order n and it is multiplied by a periodic function of log n. We now propose to prove a transfer condition of the O-type. We give the proof in some detail for two reasons: first, the implied constant in the O’s are "constructive" and tight, a fact of independent interest; second, it serves as a guiding pattern for later deriving a variety of transfer conditions. We let A 4(, n) denote the closed domain A(,n)--{z/Izl _--< +r/, IArg (z-1)1 (2.5) where we take r/> 0 and 0 < < (r/2). This domain has the form of an indented disk (a). depicted on Fig. THEOREM 1. Assume that, with the sole exception of the singularity z 1, f (z) is Univariate Algebraic Singularity Example > 0 and 0 < 49 < (r/ 2 ). Assume further A(49, r ), where analytic in the domain A that as z tends to in A, � Flajolet-Odlyzko paper from 1990: Insist that F ( z ) = O ( | 1 − z | α ) as z → 1. Also, assume that F has no singularities except for f(z)--O(ll--zl), (2.6a) z = 1 in the region pictured below: (a) (b) 4 / 33 The contour , used in the proof of Theorem 1. FIG. 1. (a) The domain A(49, ). (b
220 PHILIPPE FLAJOLET AND ANDREW ODLYZKO totic expansion as n -- , Thus the binomial coefficients (2.1), as well as their main asymptotic equivalents in (2.2), form an asymptotic scale. There is in fact a general form of (2.2). PROPOSITION 1. The binomial coefficients expressing [zn]( z) have an asymp- a{0,1,2, "’’}, (2.3) [Znl(1--Z)a’" 1+ Ia(--a kl where (2.4) e )= 2k (-1)lXk, t(a+ 1)(a+ 2)-" .(c+l) l=k with k,l_ 0 Proposition 1, although it would probably follow by close inspection of Stirling’s formula, is most easily proved by techniques introduced in 3, so that we delay the proof until then. We also observe, incidentally, that in (2.1)-(2.3 a may be complex: If c + it, we have sin (t log n) ]. [Znl(1--Z) cos (t log n) I’( -r it) In that case, the main term in (2.2), (2.3) is of order n and it is multiplied by a periodic function of log n. We now propose to prove a transfer condition of the O-type. We give the proof in some detail for two reasons: first, the implied constant in the O’s are "constructive" and tight, a fact of independent interest; second, it serves as a guiding pattern for later deriving a variety of transfer conditions. We let A 4(, n) denote the closed domain A(,n)--{z/Izl _--< +r/, IArg (z-1)1 (2.5) where we take r/> 0 and 0 < < (r/2). This domain has the form of an indented disk (a). depicted on Fig. THEOREM 1. Assume that, with the sole exception of the singularity z 1, f (z) is > 0 and 0 < 49 < (r/ 2 ). Assume further analytic in the domain A A(49, r ), where that as z tends to in A, Univariate Algebraic Singularity Example (2.6a) f(z)--O(ll--zl), � Expand C to the contour below: (a) (b) The contour , used in the proof of Theorem 1. FIG. 1. (a) The domain A(49, ). (b 5 / 33
220 PHILIPPE FLAJOLET AND ANDREW ODLYZKO totic expansion as n -- , Thus the binomial coefficients (2.1), as well as their main asymptotic equivalents in (2.2), form an asymptotic scale. There is in fact a general form of (2.2). PROPOSITION 1. The binomial coefficients expressing [zn]( z) have an asymp- a{0,1,2, "’’}, (2.3) [Znl(1--Z)a’" 1+ Ia(--a kl where (2.4) e )= 2k (-1)lXk, t(a+ 1)(a+ 2)-" .(c+l) l=k with k,l_ 0 Proposition 1, although it would probably follow by close inspection of Stirling’s formula, is most easily proved by techniques introduced in 3, so that we delay the proof until then. We also observe, incidentally, that in (2.1)-(2.3 a may be complex: If c + it, we have sin (t log n) ]. [Znl(1--Z) cos (t log n) I’( -r it) In that case, the main term in (2.2), (2.3) is of order n and it is multiplied by a periodic function of log n. We now propose to prove a transfer condition of the O-type. We give the proof in some detail for two reasons: first, the implied constant in the O’s are "constructive" and tight, a fact of independent interest; second, it serves as a guiding pattern for later deriving a variety of transfer conditions. We let A 4(, n) denote the closed domain A(,n)--{z/Izl _--< +r/, IArg (z-1)1 (2.5) where we take r/> 0 and 0 < < (r/2). This domain has the form of an indented disk (a). depicted on Fig. THEOREM 1. Assume that, with the sole exception of the singularity z 1, f (z) is > 0 and 0 < 49 < (r/ 2 ). Assume further analytic in the domain A A(49, r ), where that as z tends to in A, Univariate Algebraic Singularity Example (2.6a) f(z)--O(ll--zl), � Expand C to the contour below: (a) (b) � Analyze each part separately. The contour , used in the proof of Theorem 1. FIG. 1. (a) The domain A(49, ). (b 5 / 33
220 PHILIPPE FLAJOLET AND ANDREW ODLYZKO totic expansion as n -- , Thus the binomial coefficients (2.1), as well as their main asymptotic equivalents in (2.2), form an asymptotic scale. There is in fact a general form of (2.2). PROPOSITION 1. The binomial coefficients expressing [zn]( z) have an asymp- a{0,1,2, "’’}, (2.3) [Znl(1--Z)a’" 1+ Ia(--a kl where (2.4) e )= 2k (-1)lXk, t(a+ 1)(a+ 2)-" .(c+l) l=k with k,l_ 0 Proposition 1, although it would probably follow by close inspection of Stirling’s formula, is most easily proved by techniques introduced in 3, so that we delay the proof until then. We also observe, incidentally, that in (2.1)-(2.3 a may be complex: If c + it, we have cos (t log n) sin (t log n) ]. [Znl(1--Z) I’( -r it) In that case, the main term in (2.2), (2.3) is of order n and it is multiplied by a periodic function of log n. We now propose to prove a transfer condition of the O-type. We give the proof in some detail for two reasons: first, the implied constant in the O’s are "constructive" and tight, a fact of independent interest; second, it serves as a guiding pattern for later deriving a variety of transfer conditions. We let A 4(, n) denote the closed domain Univariate Algebraic Singularity Example A(,n)--{z/Izl _--< +r/, IArg (z-1)1 (2.5) where we take r/> 0 and 0 < < (r/2). This domain has the form of an indented disk � Since F ( z ) = O ( | 1 − z | α ), we’ll compare the integrals, (a). depicted on Fig. THEOREM 1. Assume that, with the sole exception of the singularity z 1, f (z) is > 0 and 0 < 49 < (r/ 2 ). Assume further analytic in the domain A A(49, r ), where ˆ ˆ F ( z ) z − n − 1 dz | 1 − z | α z − n − 1 dz in A, and that as z tends to C f(z)--O(ll--zl), C (2.6a) (a) (b) The contour , used in the proof of Theorem 1. FIG. 1. (a) The domain A(49, ). (b 6 / 33
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