OFFSET
2,3
COMMENTS
The array contains the coefficients for a normalized Schwarzian: Schw(g(x)) = S(f) = (x^2/6) { D^2 log(f(x)) - (1/2) [D log(f(x))]^2} with f(x)= g'(x) = 1 / [1 - c(.) x]^2 = 1 + 2 c(1) x + 3 c(2) x^2 + ....
S(f(x)) = P(2,c) x^2 + P(3,c) x^3 + P(4,c) x^4 + ..., where P(n,c) are the Neretin polynomials with an additional factor of 2.
For proof of integrality of coefficients see MathOverflow link.
Coefficients of P(n,c) sum to zero. - Tom Copeland, Jan 29 2012
REFERENCES
H. Airault, "Symmetric sums associated to the factorization of Grunsky coefficients," in Groups and Symmetries: From Neolithic Scots to John McKay, CRM Proceedings and Lecture Notes: Vol. 47, edited by J. Harnad and P. Winternitz, American Mathematical Society, p. 5, 2009.
B. Gustaffson and A. Vasil'ev, Conformal and Potential Analysis in Hele-Shaw Cells, (Advances in Mathematical Fluid Mechanics), Birkhäuser Verlag, 2006, p. 202.
LINKS
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972.
R. Hidalgo, I. Markina and A. Vasil'ev, Finite dimensional grading of the Virasoro algebra, Georg. Math. J. 14 (2007), 419-434.
I. Markina, D. Prokhorov, and A. Vasil'ev, Sub-Riemannian geometry of the coefficients of univalent functions, arXiv:math/0608532 [math.CV], p. 11, 2006.
MathOverflow, Conjecture: "Neretin polynomials" for a normalized Schwarzian have integer coefficients
V. Ovsienko and S. Tabachnikov, What is the Schwarzian Derivative?, AMS Notices 56 (01), 34-36.
A. Vasil’ev, Energy characteristics of subordination chains, arXiv:math-ph/0509072 [math-ph], p. 11, 2005.
FORMULA
See references for recurrences and lowering operators.
EXAMPLE
.. P(0,c) = 0
.. P(1,c) = 0
.. P(2,c) = c(2) - c(1)^2
.. P(3,c) = 4 c(3) - 8 c(2)c(1) + 4 c(1)^3 = 4 3' - 8 2'1' + 4 1'^3
.. P(4,c) = 10 4' - 20 3'1' - 12 2'^2 + 34 2'1'^2 - 12 1'^4
.. P(5,c) = 20 5' - 40 1'4' - 52 2'3' + 72 3'1'^2 + 84 2'^2 1'- 116 2'1'^3 + 32 1'^5
The partitions are arranged in the order of those of Abramowitz and Stegun on p. 831.
MATHEMATICA
max = 7; f[x_] := 1+Sum[(k+1)*c[k]*x^k, {k, 1, max}]; Lf[x_] := Log[f[x]]; s = (x^2/6)*(Lf''[x]-1/2*Lf'[x]^2); coes = CoefficientList[Series[s, {x, 0, max}], x]; p[n_] := coes[[n+1]]; row[n_] := Module[{r, r1, r2, r3, r4, asteg, pos}, r = List @@ Expand[p[n]]; r1 = r /. c[_] -> 1; r2 = r/r1; r3 = (r2 /. Times -> List /. c[i_]^k_ :> Array[i&, k] ) /. c[i_] :> {i}; r4 = Flatten /@ r3; asteg = Reverse /@ IntegerPartitions[n] //. {a___List, b_List, c_List, d___List} /; Length[b] > Length[c] :> {a, c, b, d}; Do[pos[i] = Position[asteg, r4[[i]], 1, 1][[1, 1]], {i, 1, Length[r]}]; Table[r1[[pos[i]]], {i, 1, Length[r]}]]; Table[row[n], {n, 2, max}] // Flatten (* Jean-François Alcover, Dec 24 2013 *)
CROSSREFS
KEYWORD
easy,sign,tabf
AUTHOR
Tom Copeland, Oct 22 2008
EXTENSIONS
Clarified relations among g(x), f(x), and Schwarzian derivative Tom Copeland, Dec 08 2009
STATUS
approved