A060628 Triangle of coefficients in expansion of elliptic function sn(u) in powers of u and k.
1, 1, 1, 1, 14, 1, 1, 135, 135, 1, 1, 1228, 5478, 1228, 1, 1, 11069, 165826, 165826, 11069, 1, 1, 99642, 4494351, 13180268, 4494351, 99642, 1, 1, 896803, 116294673, 834687179, 834687179, 116294673, 896803, 1, 1, 8071256, 2949965020, 47152124264, 109645021894, 47152124264, 2949965020, 8071256, 1
Offset: 0
Examples
sn u = u - (1 + k^2)*u^3/3! + (1 + 14*k^2 + k^4)*u^5/5! - (1 + 135*k^2 + 135*k^4 + k^6)*u^7/7! + ... The triangle T(n, m) begins: n\m 0 1 2 3 4 5 6 7 0: 1 2: 1 1 3: 1 14 1 4: 1 135 135 1 5: 1 1228 5478 1228 1 6: 1 11069 165826 165826 11069 1 7: 1 99642 4494351 13180268 4494351 99642 1 8: 1 896803 116294673 834687179 834687179 116294673 896803 1 ... reformatted. - _Wolfdieter Lang_, Jul 05 2016
References
- CRC Standard Mathematical Tables and Formulae, 30th ed. 1996, p. 526.
- I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, (5.2.24).
Links
- M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, p. 575, 16.22.1.
- A. Cayley, An Elementary Treatise on Elliptic Functions (page images), G. Bell and Sons, London, 1895, p. 56.
- F. Clarke, The Taylor Series Coefficients of the Jacobi Elliptic Functions, slides. [broken link]
- D. Dominici, Nested derivatives: A simple method for computing series expansions of inverse functions. arXiv:math/0501052v2 [math.CA], 2005.
- A. Fransen, Conjectures on the Taylor series expansion coefficients of the Jacobian elliptic function sn(n,k), Math. Comp., 37 (1981), 475-497.
- R. Fricke, Die elliptischen Funktionen und ihre Anwendungen, Erster Teil, p. 399 with p. 397.
- C. L. Mallows, Letter to N. J. A. Sloane, May 16 1973
- J. Tannery and J. Molk, Eléments de la Théorie des Fonctions Elliptiques (Vol. 4), Gauthier-Villars, Paris, 1902, p. 92.
- G. Viennot, Une interprétation combinatoire des coefficients des développements en série entière des fonctions elliptiques de Jacobi, J. Combin. Theory, A 29 (1980), 121-133.
- Eric Weisstein's World of Mathematics, Jacobi Elliptic Functions
Crossrefs
Programs
-
Maple
Maple program from Rostislav Kollman (kollman(AT)dynasig.cz), Nov 05 2009: (Start) The program generates an "all in one" triangle of Taylor coefficients of the Jacobi SN,CN,DN functions. "SN ", 1 "CN ", 1 "DN ", 1 "SN ", 1, 1 "CN ", 1, 4 "DN ", 4, 1 "SN ", 1, 14, 1 "CN ", 1, 44, 16 "DN ", 16, 44, 1 "SN ", 1, 135, 135, 1 "CN ", 1, 408, 912, 64 "DN ", 64, 912, 408, 1 "SN ", 1, 1228, 5478, 1228, 1 "CN ", 1, 3688, 30768, 15808, 256 "DN ", 256, 15808, 30768, 3688, 1 "SN ", 1, 11069, 165826, 165826, 11069, 1 "CN ", 1, 33212, 870640, 1538560, 259328, 1024 "DN ", 1024, 259328, 1538560, 870640, 33212, 1 #---------------------------------------------------------------- # Taylor series coefficients of Jacobi SN,CN,DN #---------------------------------------------------------------- n := 6: g := x: for i from 1 to 2*n do g := simplify(y*z*diff(g,x) + x*z*diff(g,y) + x*y*diff(g,z)); if(type(i,odd))then SN := simplify(sort(subs(z = k,subs(y = 1,subs(x = 0,g)))) / k); # lprint("SN ",SN); lprint("SN ",seq(coeff(SN, k, j),j=0..i-1,2)); else CN := simplify(sort(subs(z = 1,subs(y = 0,subs(x = k,g)))) / k); DN := simplify(sort(subs(z = 0,subs(y = k,subs(x = 1,g))))); # lprint("CN ",CN); # lprint("DN ",DN); lprint("CN ",seq(coeff(CN, k, j),j=0..i-2,2)); lprint("DN ",seq(coeff(DN, k, j),j=2..i,2)); end; end: (End) A060628 := proc(n,m) JacobiSN(z,k) ; coeftayl(%,z=0,2*n+1) ; (-1)^n*coeftayl(%,k=0,2*m)*(2*n+1)! ; end proc: # alternative program, R. J. Mathar, Jan 30 2011 -
Mathematica
maxn = 8; se = Series[ JacobiSN[u, m], {u, 0, 2*maxn + 1 }]; cc = Partition[ CoefficientList[se, u], 2][[All, 2]]; Flatten[ (CoefficientList[#, m] & /@ cc)* Table[(-1)^n*(2*n + 1)!, {n, 0, maxn}]] (* Jean-François Alcover, Sep 21 2011 *)
Formula
Sum_{n>=0} Sum_{k=0..n} (-1)^n*T(n, k)*y^(2*k)*x^(2*n+1)/(2*n+1)! = JacobiSN(x, y).
JacobiSN(x, y) = 1*x + (-1/6 - (1/6)*y^2)*x^3 + (1/120 + (7/60)*y^2 + (1/120)*y^4)*x^5 + (-1/5040 - (3/112)*y^4 - (3/112)*y^2 - (1/5040)*y^6)*x^7 + (1/362880 + (307/90720)*y^6 + (913/60480)*y^4 + (307/90720)*y^2 + (1/362880)*y^8)*x^9 + O(x^11).
From Peter Bala, Aug 23 2011: (Start)
Let f(x) = sqrt((1-x^2)*(1-k^2*x^2)).
Define the nested derivative D^n[f](x) by means of the recursion D^0[f](x) = 1 and D^(n+1)[f](x) = d/dx(f(x)*D^n[f](x)) for n >= 0.
Then the coefficient polynomial R(2*n+1,k) of u^(2*n+1)/(2*n+1)! is given by R(2*n+1,k) = D^(2*n)[f](0) - apply [Dominici, Theorem 4.1].
See A145271 for the coefficients in the expansion of D^n[f](x) in powers of f(x).
(End)
sn(u|k^2) = Sum_{n>=0} a_n(k^2)*u^(2*n+1)/(2*n+1)!. For the recurrence of the row polynomials a_n(k^2) = Sum_{m=0..n} (-1)^n*T(n, m)*k^(2*m), see the Fricke reference. - Wolfdieter Lang, Jul 05 2016