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A324609
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E.g.f. C(x,y) = cosh( Integral C(x,y)*C(y,x) dx ), where C(y,x) = cosh( Integral C(x,y)*C(y,x) dy ).
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5
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1, 1, 0, 5, 2, 0, 61, 28, 16, 0, 1385, 662, 440, 272, 0, 50521, 24568, 17176, 12448, 7936, 0, 2702765, 1326122, 949520, 727232, 546560, 353792, 0, 199360981, 98329108, 71350336, 56140288, 44720896, 34259968, 22368256, 0, 19391512145, 9596075582, 7020926600, 5610570992, 4600173440, 3742967552, 2900372480, 1903757312, 0, 2404879675441, 1192744081648, 877465887496, 708137588128, 590470281856, 495154244608, 408133590016, 318605529088, 209865342976, 0
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OFFSET
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0,4
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COMMENTS
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Related identity: (1 + sin(z))/cos(z) = exp( Integral 1/cos(z) dz ).
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LINKS
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FORMULA
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E.g.f. Cx = C(x,y) and related functions Sx = S(x,y), Cy = C(y,x), and Sy = S(y,x) satisfy the following relations.
(1a) Cx = 1 + Integral Sx * Cx*Cy dx.
(1b) Sx = Integral Cx * Cx*Cy dx.
(1c) Cy = 1 + Integral Sy * Cx*Cy dy.
(1d) Sy = Integral Cy * Cx*Cy dy.
(2a) Cx^2 - Sx^2 = 1.
(2b) Cy^2 - Sy^2 = 1.
(3a) Cx = cosh( Integral Cx*Cy dx ).
(3b) Sx = sinh( Integral Cx*Cy dx ).
(3c) Cy = cosh( Integral Cx*Cy dy ).
(3d) Sy = sinh( Integral Cx*Cy dy ).
(4a) Cx + Sx = exp( Integral Cx*Cy dx ).
(4b) Cy + Sy = exp( Integral Cx*Cy dy ).
(5a) (Cx + Sx)*(Cy + Sy) = (1 + sin(x+y))/cos(x+y).
(5b) (Cx + Sx)*(Cy - Sy) = (1 + sin(x-y))/cos(x-y).
(6a) Cx*Cy + Sx*Sy = 1/cos(x+y).
(6b) Cx*Sy + Sx*Cy = tan(x+y).
(7a) Cx*Cy = ( 1/cos(x+y) + 1/cos(x-y) )/2.
(7b) Sx*Sy = ( 1/cos(x+y) - 1/cos(x-y) )/2.
(7c) Cx*Sy = ( tan(x+y) - tan(x-y) )/2.
(7d) Sx*Cy = ( tan(x+y) + tan(x-y) )/2.
(8a) Cx*Cy = cos(x)*cos(y) / (cos(x+y)*cos(x-y)).
(8b) Sx*Sy = sin(x)*sin(y) / (cos(x+y)*cos(x-y)).
(8c) Cx*Sy = cos(y)*sin(y) / (cos(x+y)*cos(x-y)).
(8d) Sx*Cy = sin(x)*cos(x) / (cos(x+y)*cos(x-y)).
(9a) Cx + Sx = sqrt( (1 + sin(x+y))/cos(x+y) * (1 + sin(x-y))/cos(x-y) ).
(9b) Cy + Sy = sqrt( (1 + sin(x+y))/cos(x+y) * (1 - sin(x-y))/cos(x-y) ).
(9c) Cx - Sx = sqrt( (1 - sin(x+y))/cos(x+y) * (1 - sin(x-y))/cos(x-y) ).
(9d) Cy - Sy = sqrt( (1 - sin(x+y))/cos(x+y) * (1 + sin(x-y))/cos(x-y) ).
Let E(x,y) = sqrt( (1 + sin(x+y))/cos(x+y) * (1 + sin(x-y))/cos(x-y) ) then
(10a) E(x,y) = C(x,y) + S(x,y) where E(-x,y) = 1/E(x,y),
(10b) C(x,y) = (E(x,y) + E(-x,y))/2,
(10c) S(x,y) = (E(x,y) - E(-x,y))/2.
PARTICULAR ARGUMENTS.
E.g.f. at y = 0: C(x,y=0) = 1/cos(x).
E.g.f. at y = x: C(x,y=x) = cos(x)/sqrt(cos(2*x)).
FORMULAS INVOLVING TERMS.
T(n,n-1) = A000182(n) for n >= 1, where A000182 is the tangent numbers.
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EXAMPLE
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E.g.f.: C(x,y) = 1 + (1*x^2/2!) + (5*x^4/4! + 2*x^2*y^2/(2!*2!)) + (61*x^6/6! + 28*x^4*y^2/(4!*2!) + 16*x^2*y^4/(2!*4!)) + (1385*x^8/8! + 662*x^6*y^2/(6!*2!) + 440*x^4*y^4/(4!*4!) + 272*x^2*y^6/(2!*6!)) + (50521*x^10/10! + 24568*x^8*y^2/(8!*2!) + 17176*x^6*y^4/(6!*4!) + 12448*x^4*y^6/(4!*6!) + 7936*x^2*y^8/(2!*8!)) + (2702765*x^12/12! + 1326122*x^10*y^2/(10!*2!) + 949520*x^8*y^4/(8!*4!) + 727232*x^6*y^6/(6!*6!) + 546560*x^4*y^8/(4!*8!) + 353792*x^2*y^10/(2!*10!)) + ...
such that C(x,y) = cosh( Integral C(x,y)*C(y,x) dx ).
Explicitly,
C(x,y) = ( sqrt( (1 + sin(x+y))/cos(x+y) * (1 + sin(x-y))/cos(x-y) ) + sqrt( (1 - sin(x+y))/cos(x+y) * (1 - sin(x-y))/cos(x-y) ) )/2.
This triangle of coefficients T(n,k) of x^(2*n-2*k)*y^(2*k)/((2*n-2*k)!*(2*k)!) in e.g.f. C(x,y) begins
1;
1, 0;
5, 2, 0;
61, 28, 16, 0;
1385, 662, 440, 272, 0;
50521, 24568, 17176, 12448, 7936, 0;
2702765, 1326122, 949520, 727232, 546560, 353792, 0;
199360981, 98329108, 71350336, 56140288, 44720896, 34259968, 22368256, 0;
19391512145, 9596075582, 7020926600, 5610570992, 4600173440, 3742967552, 2900372480, 1903757312, 0;
2404879675441, 1192744081648, 877465887496, 708137588128, 590470281856, 495154244608, 408133590016, 318605529088, 209865342976, 0; ...
RELATED SERIES.
C(y,x) = 1 + (1*y^2/2!) + (2*x^2*y^2/(2!*2!) + 5*y^4/4!) + (16*x^4*y^2/(4!*2!) + 28*x^2*y^4/(2!*4!) + 61*y^6/6!) + (272*x^6*y^2/(6!*2!) + 440*x^4*y^4/(4!*4!) + 662*x^2*y^6/(2!*6!) + 1385*y^8/8!) + (7936*x^8*y^2/(8!*2!) + 12448*x^6*y^4/(6!*4!) + 17176*x^4*y^6/(4!*6!) + 24568*x^2*y^8/(2!*8!) + 50521*y^10/10!) + (353792*x^10*y^2/(10!*2!) + 546560*x^8*y^4/(8!*4!) + 727232*x^6*y^6/(6!*6!) + 949520*x^4*y^8/(4!*8!) + 1326122*x^2*y^10/(2!*10!) + 2702765*y^12/12!) + ...
such that C(y,x) = cosh( Integral C(x,y)*C(y,x) dy ).
S(x,y) = x + (2*x^3/3! + 1*x*y^2/2!) + (16*x^5/5! + 8*x^3*y^2/(3!*2!) + 5*x*y^4/4!) + (272*x^7/7! + 136*x^5*y^2/(5!*2!) + 94*x^3*y^4/(3!*4!) + 61*x*y^6/6!) + (7936*x^9/9! + 3968*x^7*y^2/(7!*2!) + 2840*x^5*y^4/(5!*4!) + 2108*x^3*y^6/(3!*6!) + 1385*x*y^8/8!) + (353792*x^11/11! + 176896*x^9*y^2/(9!*2!) + 128704*x^7*y^4/(7!*4!) + 100096*x^5*y^6/(5!*6!) + 76474*x^3*y^8/(3!*8!) + 50521*x*y^10/10!) + (22368256*x^13/13! + 11184128*x^11*y^2/(11!*2!) + 8211200*x^9*y^4/(9!*4!) + 6531968*x^7*y^6/(7!*6!) + 5261120*x^5*y^8/(5!*8!) + 4079408*x^3*y^10/(3!*10!) + 2702765*x*y^12/12!) + ...
such that C(x,y)^2 - S(x,y)^2 = 1.
S(y,x) = y + (1*x^2*y/2! + 2*y^3/3!) + (5*x^4*y/4! + 8*x^2*y^3/(2!*3!) + 16*y^5/5!) + (61*x^6*y/6! + 94*x^4*y^3/(4!*3!) + 136*x^2*y^5/(2!*5!) + 272*y^7/7!) + (1385*x^8*y/8! + 2108*x^6*y^3/(6!*3!) + 2840*x^4*y^5/(4!*5!) + 3968*x^2*y^7/(2!*7!) + 7936*y^9/9!) + (50521*x^10*y/10! + 76474*x^8*y^3/(8!*3!) + 100096*x^6*y^5/(6!*5!) + 128704*x^4*y^7/(4!*7!) + 176896*x^2*y^9/(2!*9!) + 353792*y^11/11!) + (2702765*x^12*y/12! + 4079408*x^10*y^3/(10!*3!) + 5261120*x^8*y^5/(8!*5!) + 6531968*x^6*y^7/(6!*7!) + 8211200*x^4*y^9/(4!*9!) + 11184128*x^2*y^11/(2!*11!) + 22368256*y^13/13!) + ...
such that C(y,x)^2 - S(y,x)^2 = 1.
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PROG
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(PARI) {T(n, k) = my(Cx = 1 + x*O(x^(2*n)), Cy = 1 + y*O(y^(2*n)));
for(i=1, 2*n,
Cx = cosh(intformal(Cx*Cy, x));
Cy = cosh(intformal(Cx*Cy, y)); );
Sx = sinh(intformal(Cx*Cy, x));
Sy = sinh(intformal(Cx*Cy, y));
(2*n-2*k)!*(2*k)! * polcoeff(polcoeff(Cx, 2*n-2*k, x), 2*k, y)}
for(n=0, 10, for(k=0, n, print1( T(n, k), ", ")); print(""))
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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