OFFSET
0,5
COMMENTS
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FORMULA
The special functions S(x,y), C(x,y), and D(x,y) satisfy the following relations.
(1a) S(x,y) = Integral C(x,y) * C(y,x) dx.
(1b) S(y,x) = Integral C(y,x) * C(x,y) dy.
(1c) C(x,y) = 1 + Integral S(x,y) * C(y,x) dx.
(1d) C(y,x) = 1 + Integral S(y,x) * C(x,y) dy.
(2a) C(x,y)^2 - S(x,y)^2 = 1.
(2b) C(y,x)^2 - S(y,x)^2 = 1.
(3a) S(x,y) = sinh( Integral C(y,x) dx ).
(3b) S(y,x) = sinh( Integral C(x,y) dy ).
(3c) C(x,y) = cosh( Integral C(y,x) dx ).
(3d) C(y,x) = cosh( Integral C(x,y) dy ).
(4a) C(x,y) + S(x,y) = exp( Integral C(y,x) dx ).
(4b) C(y,x) + S(y,x) = exp( Integral C(x,y) dy ).
(5a) d/dx S(x,y) = C(x,y) * C(y,x).
(5b) d/dx C(x,y) = S(x,y) * C(y,x).
(5c) d/dy S(y,x) = C(y,x) * C(x,y).
(5d) d/dy C(y,x) = S(y,x) * C(x,y).
Introducing function D(x,y) completes the symmetric relations as follows.
(6a) D(x,y) = Integral S(y,x) * C(x,y) dx.
(6b) D(y,x) = Integral S(x,y) * C(y,x) dy.
(7a) S(x,y) = sinh(x) + Integral C(x,y) * D(x,y) dy.
(7b) S(y,x) = sinh(y) + Integral C(y,x) * D(y,x) dx.
(7c) C(x,y) = cosh(x) + Integral S(x,y) * D(x,y) dy.
(7d) C(y,x) = cosh(y) + Integral S(y,x) * D(y,x) dx.
(8a) C(x,y) + S(x,y) = exp( x + Integral D(x,y) dy ).
(8b) C(y,x) + S(y,x) = exp( y + Integral D(y,x) dx ).
(9a) Integral C(y,x) dx = x + Integral D(x,y) dy.
(9b) Integral C(x,y) dy = y + Integral D(y,x) dx.
(10a) d/dy S(x,y) = C(x,y) * D(x,y).
(10b) d/dy C(x,y) = S(x,y) * D(x,y).
(10c) d/dx S(y,x) = C(y,x) * D(y,x).
(10d) d/dx C(y,x) = S(y,x) * D(y,x).
(10e) d/dx D(x,y) = S(y,x) * C(x,y).
(10f) d/dy D(y,x) = S(x,y) * C(y,x).
For brevity, let Cx = C(x,y), Cy = C(y,x), Sx = S(x,y), Sy = S(y,x), Dx = D(x,y), Dy = D(y,x), then further relations may be written as follows.
(11a) Cx*Cy + Sx*Sy = cosh(y) + Integral (Cy + Dy)*(Sx*Cy + Cx*Sy) dx.
(11b) Sx*Cy + Cx*Sy = sinh(y) + Integral (Cy + Dy)*(Cx*Cy + Sx*Sy) dx.
(11c) Cx*Cy + Sx*Sy = cosh(x) + Integral (Cx + Dx)*(Sx*Cy + Cx*Sy) dy.
(11d) Sx*Cy + Cx*Sy = sinh(x) + Integral (Cx + Dx)*(Cx*Cy + Sx*Sy) dy.
(12a) (Cx + Sx)*(Cy + Sy) = exp( y + Integral Cy + Dy dx ).
(12b) (Cx + Sx)*(Cy + Sy) = exp( x + Integral Cx + Dx dy ).
(12c) (Cx + Sx)*(Cy + Sy) = exp( x + y + Integral Dx dy + Integral Dy dx ).
(12d) (Cx + Sx)*(Cy + Sy) = exp( x + y + Integral Integral Sx*Cy + Cx*Sy dx dy ).
(12e) x + Integral (Cx + Dx) dy = y + Integral (Cy + Dy) dx.
(13a) d/dx (Cx + Sx)*(Cy + Sy) = (Cx + Sx)*(Cy + Sy)*(Cy + Dy).
(13b) d/dy (Cx + Sx)*(Cy + Sy) = (Cx + Sx)*(Cy + Sy)*(Cx + Dx).
(14a) (Cx + Sx)*(Cy + Sy) = exp(y) + Integral (Cx + Sx)*(Cy + Sy)*(Cy + Dy) dx.
(14b) (Cx + Sx)*(Cy + Sy) = exp(x) + Integral (Cx + Sx)*(Cy + Sy)*(Cx + Dx) dy.
EXAMPLE
E.g.f. C(y,x) = Sum_{n>=0} Sum_{k=0..n} T(n,k) * x^(2*n-2*k)*y^(2*k)/(2*n)! begins
C(y,x) = 1 + (1*y^2)/2! + (12*x^2*y^2 + 1*y^4)/4! + (120*x^4*y^2 + 180*x^2*y^4 + 1*y^6)/6! + (896*x^6*y^2 + 9520*x^4*y^4 + 2632*x^2*y^6 + 1*y^8)/8! + (5760*x^8*y^2 + 369600*x^6*y^4 + 504000*x^4*y^6 + 37080*x^2*y^8 + 1*y^10)/10! + (33792*x^10*y^2 + 12735360*x^8*y^4 + 57376704*x^6*y^6 + 23562000*x^4*y^8 + 487476*x^2*y^10 + 1*y^12)/12! + (186368*x^12*y^2 + 399246848*x^10*y^4 + 5256451200*x^8*y^6 + 6645999360*x^6*y^8 + 1039654616*x^4*y^10 + 6045676*x^2*y^12 + 1*y^14)/14! + (983040*x^14*y^2 + 11498905600*x^12*y^4 + 427859367936*x^10*y^6 + 1336544352000*x^8*y^8 + 674286412800*x^6*y^10 + 44074736160*x^4*y^12 + 71745360*x^2*y^14 + 1*y^16)/16! + ...
This series may be defined by
C(y,x) = cosh( Integral C(x,y) dy ), and
C(x,y) = cosh( Integral C(y,x) dx ).
TRIANGLE.
This triangle of coefficients T(n,k) of x^(2*n-2*k)*y^(2*k)/(2*n)! in C(y,x) starts
1;
0, 1;
0, 12, 1;
0, 120, 180, 1;
0, 896, 9520, 2632, 1;
0, 5760, 369600, 504000, 37080, 1;
0, 33792, 12735360, 57376704, 23562000, 487476, 1;
0, 186368, 399246848, 5256451200, 6645999360, 1039654616, 6045676, 1;
0, 983040, 11498905600, 427859367936, 1336544352000, 674286412800, 44074736160, 71745360, 1;
0, 5013504, 308405698560, 32219490680832, 224718683904000, 271641150443520, 63624274826112, 1793634471360, 823269744, 1; ...
PROG
(PARI) {T(n, k) = my(Sx=x, Sy=y, Cx=1, Cy=1); for(i=1, 2*n,
Sx = intformal( Cx*Cy +x*O(x^(2*n)), x);
Cx = 1 + intformal( Sx*Cy, x);
Sy = intformal( Cy*Cx +y*O(y^(2*k)), y);
Cy = 1 + intformal( Sy*Cx, y));
(2*n)! *polcoeff(polcoeff(Cy, 2*n-2*k, x), 2*k, y)}
for(n=0, 10, for(k=0, n, print1( T(n, k), ", ")); print(""))
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Jan 01 2019
STATUS
approved