Abstract

A recently introduced method for characterizing the shape of rotationally symmetric aspheres is generalized here for application to a wide class of freeform optics. New sets of orthogonal polynomials are introduced along with robust and efficient algorithms for computing the surface shape as well as its derivatives of any order. By construction, the associated characterization offers a rough interpretation of shape at a glance and it facilitates a range of estimates of manufacturability.

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References

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  1. H. J. Birchall, “Lenses and their combination and arrangement in various instruments and apparatus,” U.S. patent 2,001,952 (21 May 1935).
  2. H. J. Birchall, “Lens of variable focal power having surfaces of involute form,” U.S. patent 2,475,275 (7 March 1949).
  3. C. W. Kanolt, “Multifocal ophthalmic lenses,” U.S. patent 2,878,721 (24 March 1959).
  4. W. T. Plummer, J. G. Baker, and J. Van Tassell, “Photographic optical systems with nonrotational aspheric surfaces,” Appl. Opt. 38(16), 3572–3592 (1999).
    [CrossRef] [PubMed]
  5. L. Wang, P. Benítez, J. C. Miñano, J. Infante, and G. Biot, “Advances in the SMS design method for imaging optics,” Proc. SPIE 8167, 81670M (2011).
    [CrossRef]
  6. F. Muñoz, P. Benítez, and J. C. Miñano, “High-order aspherics: the SMS nonimaging design method applied to imaging optics,” Proc. SPIE 7061, 70610G, 70610G-9 (2008).
    [CrossRef]
  7. K. H. Fuerschbach, K. P. Thompson, and J. P. Rolland, “A new generation of optical systems with φ-polynomial surfaces,” Proc. SPIE 7652, 76520C, 76520C-7 (2010).
    [CrossRef]
  8. J. R. Rogers, “A comparison of anamorphic, keystone, and Zernike surface types for aberration correction,” Proc. SPIE 7652, 76520B, 76520B-8 (2010).
    [CrossRef]
  9. A. Yabe, “Method to allocate freeform surfaces in axially asymmetric optical systems,” Proc. SPIE 8167, 816703, 816703-10 (2011).
    [CrossRef]
  10. R. Steinkopf, L. Dick, T. Kopf, A. Gebhardt, S. Risse, and R. Eberhardt, “Data handling and representation of freeform surfaces,” Proc. SPIE 8169, 81690X, 81690X-9 (2011).
    [CrossRef]
  11. P. Jester, C. Menke, and K. Urban, “B-spline representation of optical surfaces and its accuracy in a ray trace algorithm,” Appl. Opt. 50(6), 822–828 (2011).
    [CrossRef] [PubMed]
  12. G. W. Forbes, “Manufacturability estimates for optical aspheres,” Opt. Express 19(10), 9923–9941 (2011).
    [CrossRef] [PubMed]
  13. G. W. Forbes, “Shape specification for axially symmetric optical surfaces,” Opt. Express 15(8), 5218–5226 (2007).
    [CrossRef] [PubMed]
  14. G. W. Forbes, “Robust, efficient computational methods for axially symmetric optical aspheres,” Opt. Express 18(19), 19700–19712 (2010).
    [CrossRef] [PubMed]
  15. C. Zhao and J. H. Burge, “Orthonormal vector polynomials in a unit circle, Part I: Basis set derived from gradients of Zernike polynomials,” Opt. Express 15(26), 18014–18024 (2007).
    [CrossRef] [PubMed]
  16. G. W. Forbes, “Robust and fast computation for the polynomials of optics,” Opt. Express 18(13), 13851–13862 (2010).
    [CrossRef] [PubMed]

2011

L. Wang, P. Benítez, J. C. Miñano, J. Infante, and G. Biot, “Advances in the SMS design method for imaging optics,” Proc. SPIE 8167, 81670M (2011).
[CrossRef]

A. Yabe, “Method to allocate freeform surfaces in axially asymmetric optical systems,” Proc. SPIE 8167, 816703, 816703-10 (2011).
[CrossRef]

R. Steinkopf, L. Dick, T. Kopf, A. Gebhardt, S. Risse, and R. Eberhardt, “Data handling and representation of freeform surfaces,” Proc. SPIE 8169, 81690X, 81690X-9 (2011).
[CrossRef]

P. Jester, C. Menke, and K. Urban, “B-spline representation of optical surfaces and its accuracy in a ray trace algorithm,” Appl. Opt. 50(6), 822–828 (2011).
[CrossRef] [PubMed]

G. W. Forbes, “Manufacturability estimates for optical aspheres,” Opt. Express 19(10), 9923–9941 (2011).
[CrossRef] [PubMed]

2010

K. H. Fuerschbach, K. P. Thompson, and J. P. Rolland, “A new generation of optical systems with φ-polynomial surfaces,” Proc. SPIE 7652, 76520C, 76520C-7 (2010).
[CrossRef]

J. R. Rogers, “A comparison of anamorphic, keystone, and Zernike surface types for aberration correction,” Proc. SPIE 7652, 76520B, 76520B-8 (2010).
[CrossRef]

G. W. Forbes, “Robust, efficient computational methods for axially symmetric optical aspheres,” Opt. Express 18(19), 19700–19712 (2010).
[CrossRef] [PubMed]

G. W. Forbes, “Robust and fast computation for the polynomials of optics,” Opt. Express 18(13), 13851–13862 (2010).
[CrossRef] [PubMed]

2008

F. Muñoz, P. Benítez, and J. C. Miñano, “High-order aspherics: the SMS nonimaging design method applied to imaging optics,” Proc. SPIE 7061, 70610G, 70610G-9 (2008).
[CrossRef]

2007

1999

Baker, J. G.

Benítez, P.

L. Wang, P. Benítez, J. C. Miñano, J. Infante, and G. Biot, “Advances in the SMS design method for imaging optics,” Proc. SPIE 8167, 81670M (2011).
[CrossRef]

F. Muñoz, P. Benítez, and J. C. Miñano, “High-order aspherics: the SMS nonimaging design method applied to imaging optics,” Proc. SPIE 7061, 70610G, 70610G-9 (2008).
[CrossRef]

Biot, G.

L. Wang, P. Benítez, J. C. Miñano, J. Infante, and G. Biot, “Advances in the SMS design method for imaging optics,” Proc. SPIE 8167, 81670M (2011).
[CrossRef]

Burge, J. H.

Dick, L.

R. Steinkopf, L. Dick, T. Kopf, A. Gebhardt, S. Risse, and R. Eberhardt, “Data handling and representation of freeform surfaces,” Proc. SPIE 8169, 81690X, 81690X-9 (2011).
[CrossRef]

Eberhardt, R.

R. Steinkopf, L. Dick, T. Kopf, A. Gebhardt, S. Risse, and R. Eberhardt, “Data handling and representation of freeform surfaces,” Proc. SPIE 8169, 81690X, 81690X-9 (2011).
[CrossRef]

Forbes, G. W.

Fuerschbach, K. H.

K. H. Fuerschbach, K. P. Thompson, and J. P. Rolland, “A new generation of optical systems with φ-polynomial surfaces,” Proc. SPIE 7652, 76520C, 76520C-7 (2010).
[CrossRef]

Gebhardt, A.

R. Steinkopf, L. Dick, T. Kopf, A. Gebhardt, S. Risse, and R. Eberhardt, “Data handling and representation of freeform surfaces,” Proc. SPIE 8169, 81690X, 81690X-9 (2011).
[CrossRef]

Infante, J.

L. Wang, P. Benítez, J. C. Miñano, J. Infante, and G. Biot, “Advances in the SMS design method for imaging optics,” Proc. SPIE 8167, 81670M (2011).
[CrossRef]

Jester, P.

Kopf, T.

R. Steinkopf, L. Dick, T. Kopf, A. Gebhardt, S. Risse, and R. Eberhardt, “Data handling and representation of freeform surfaces,” Proc. SPIE 8169, 81690X, 81690X-9 (2011).
[CrossRef]

Menke, C.

Miñano, J. C.

L. Wang, P. Benítez, J. C. Miñano, J. Infante, and G. Biot, “Advances in the SMS design method for imaging optics,” Proc. SPIE 8167, 81670M (2011).
[CrossRef]

F. Muñoz, P. Benítez, and J. C. Miñano, “High-order aspherics: the SMS nonimaging design method applied to imaging optics,” Proc. SPIE 7061, 70610G, 70610G-9 (2008).
[CrossRef]

Muñoz, F.

F. Muñoz, P. Benítez, and J. C. Miñano, “High-order aspherics: the SMS nonimaging design method applied to imaging optics,” Proc. SPIE 7061, 70610G, 70610G-9 (2008).
[CrossRef]

Plummer, W. T.

Risse, S.

R. Steinkopf, L. Dick, T. Kopf, A. Gebhardt, S. Risse, and R. Eberhardt, “Data handling and representation of freeform surfaces,” Proc. SPIE 8169, 81690X, 81690X-9 (2011).
[CrossRef]

Rogers, J. R.

J. R. Rogers, “A comparison of anamorphic, keystone, and Zernike surface types for aberration correction,” Proc. SPIE 7652, 76520B, 76520B-8 (2010).
[CrossRef]

Rolland, J. P.

K. H. Fuerschbach, K. P. Thompson, and J. P. Rolland, “A new generation of optical systems with φ-polynomial surfaces,” Proc. SPIE 7652, 76520C, 76520C-7 (2010).
[CrossRef]

Steinkopf, R.

R. Steinkopf, L. Dick, T. Kopf, A. Gebhardt, S. Risse, and R. Eberhardt, “Data handling and representation of freeform surfaces,” Proc. SPIE 8169, 81690X, 81690X-9 (2011).
[CrossRef]

Thompson, K. P.

K. H. Fuerschbach, K. P. Thompson, and J. P. Rolland, “A new generation of optical systems with φ-polynomial surfaces,” Proc. SPIE 7652, 76520C, 76520C-7 (2010).
[CrossRef]

Urban, K.

Van Tassell, J.

Wang, L.

L. Wang, P. Benítez, J. C. Miñano, J. Infante, and G. Biot, “Advances in the SMS design method for imaging optics,” Proc. SPIE 8167, 81670M (2011).
[CrossRef]

Yabe, A.

A. Yabe, “Method to allocate freeform surfaces in axially asymmetric optical systems,” Proc. SPIE 8167, 816703, 816703-10 (2011).
[CrossRef]

Zhao, C.

Appl. Opt.

Opt. Express

Proc. SPIE

L. Wang, P. Benítez, J. C. Miñano, J. Infante, and G. Biot, “Advances in the SMS design method for imaging optics,” Proc. SPIE 8167, 81670M (2011).
[CrossRef]

F. Muñoz, P. Benítez, and J. C. Miñano, “High-order aspherics: the SMS nonimaging design method applied to imaging optics,” Proc. SPIE 7061, 70610G, 70610G-9 (2008).
[CrossRef]

K. H. Fuerschbach, K. P. Thompson, and J. P. Rolland, “A new generation of optical systems with φ-polynomial surfaces,” Proc. SPIE 7652, 76520C, 76520C-7 (2010).
[CrossRef]

J. R. Rogers, “A comparison of anamorphic, keystone, and Zernike surface types for aberration correction,” Proc. SPIE 7652, 76520B, 76520B-8 (2010).
[CrossRef]

A. Yabe, “Method to allocate freeform surfaces in axially asymmetric optical systems,” Proc. SPIE 8167, 816703, 816703-10 (2011).
[CrossRef]

R. Steinkopf, L. Dick, T. Kopf, A. Gebhardt, S. Risse, and R. Eberhardt, “Data handling and representation of freeform surfaces,” Proc. SPIE 8169, 81690X, 81690X-9 (2011).
[CrossRef]

Other

H. J. Birchall, “Lenses and their combination and arrangement in various instruments and apparatus,” U.S. patent 2,001,952 (21 May 1935).

H. J. Birchall, “Lens of variable focal power having surfaces of involute form,” U.S. patent 2,475,275 (7 March 1949).

C. W. Kanolt, “Multifocal ophthalmic lenses,” U.S. patent 2,878,721 (24 March 1959).

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Figures (8)

Fig. 1
Fig. 1

An off-axis section of a paraboloid is commonly encountered and has some of the challenges of more general freeform optics. The coordinate origin used for Eq. (2.1) is neither at the top nor the bottom of the cylinder as drawn for clarity at left, but in a plane that is tangent to the part’s vertex as shown at right. The separation between the cylinder and part axes is s.

Fig. 2
Fig. 2

The characterization of a segment of a general freeform surface is depicted at left. In this case, as may be familiar from segmented mirrors, the segment of interest is taken to be the intersection of a hexagonal column with the surface. I opt for the coordinate origin to sit on the surface and choose the green cylinder to tightly enclose the segment of interest on the surface. The cylinder will typically be chosen to be nominally normal to the surface segment, as shown. The case for a part that resembles the paraboloidal segment of Fig. 1 is depicted at right.

Fig. 3
Fig. 3

A sample of the polynomials constructed to be orthogonal in gradient. The columns for each azimuthal order contain the associated polynomials for n = 0, 1, 2,...

Fig. 4
Fig. 4

Plots of a sample of the basis members tabulated in Fig. 3. The four paired rows correspond to m = 0, 1, 2, and 5 while the five columns are for n = 0, 1, 2, 3, and 4. The upper row of each pair plots the associated basis function while the lower shows the modulus of their gradient. Importantly, the color scale is the same for all of the contour maps and it spans from 0 (purple) to 2 (red); a gradient of unity corresponds to the green of the plot for m = 1 and n = 0.

Fig. 5
Fig. 5

Plots of the gradient field of three of the family of functions plotted in Fig. 4. It is the point-by-point inner products of different members of such maps that integrate to zero. The scale is fixed in all of these plots so that an arrow whose length matches the grid spacing represents a gradient of magnitude equal to the square root of two.

Fig. 6
Fig. 6

Fitted coefficients in units of nm. These 23 coefficients are truncated as discussed in the paragraph before Eq. (2.3) with T = 8. If the part did not have a plane of symmetry, b n m with m > 1 would bring the number of coefficients up to 43. These values have been rounded to the nearest nm and the results were used in computing the fit error of Fig. 7.

Fig. 7
Fig. 7

The sample paraboloid is shown at bottom along with the cylinder that encloses the off-axis section of interest and the associated best-fit sphere in red. The sag departure is plotted in the upper row along with the difference between this sag and that associated with the coefficients of Fig. 6. Notice that the fit error remains less than a nanometer.

Fig. 8
Fig. 8

The corresponding coefficients for the case of Fig. 6, but where the cylinder has been tilted by 20.2223 mrad to enforce Eq. (2.3). That is, as highlighted above in red, a 0 1 now vanishes. The inverse of the best-fit curvature for this case is 37.432729 mm.

Equations (57)

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z= c con [ (x+s) 2 + y 2 ] 1+ 1(1+κ) c con 2 [ (x+s) 2 + y 2 ] = c con ( ρ 2 +2sρcosθ+ s 2 ) 1+ 1(1+κ) c con 2 ( ρ 2 +2sρcosθ+ s 2 ) ,
f(ρ,θ)= c ρ 2 1+ 1 c 2 ρ 2 + 1 1 c 2 ρ 2 { u 2 (1 u 2 ) n=0 N a n 0 Q n 0 ( u 2 ) + m=1 M u m n=0 N [ a n m cos(mθ)+ b n m sin(mθ)] Q n m ( u 2 ) }.
a 0 1 = b 0 1 =0.
n=0 N a n 1 Q n 1 (1) = n=0 N b n 1 Q n 1 (1) =0,and n=0 N a n 1 Q n 1 (0) = n=0 N b n 1 Q n 1 (0) =0.
δ(u,θ):= u 2 (1 u 2 ) n=0 N a n 0 Q n 0 ( u 2 ) + m=1 M u m n=0 N [ a n m cos(mθ)+ b n m sin(mθ)] Q n m ( u 2 ) ,
| δ( u,θ ) | 2 = ( δ u ) 2 + 1 u 2 ( δ θ ) 2 = m,n [ ( a n m ) 2 + ( b n m ) 2 ] .
g(u,θ) := 0 1 π π g(u,θ)w(u)udθdu / 0 1 π π w(u)udθdu .
g(u,θ) := 1 π 2 0 1 π π g(u,θ) [1 u 2 ] 1/2 dθdu .
1 π 0 1 { u [ u m Q n m ( u 2 )],m u m1 Q n m ( u 2 ) }{ u [ u m Q n m ( u 2 )],m u m1 Q n m ( u 2 ) } du 1 u 2
a n m cos(mθ)+ b n m sin(mθ)= α n m cos(mθ ϕ n m ).
| δ(u,θ) | 2 = m,n [ ( a n m ) 2 + ( b n m ) 2 ] = m,n ( α n m ) 2 .
Q 0 6 (x)= 8 9 7 , Q 1 6 (x)= 8 9 1397 (7772x),
Q 0 7 (x)= 32 7 231 , Q 0 8 (x)= 8 858 .
z= c con ( ρ 2 +2sρcosθ+ s 2 ) 1+ 1(1+κ) c con 2 ( ρ 2 +2sρcosθ+ s 2 ) + δ(ρ/ ρ max ,θ) σ(ρ,θ) ,
σ(ρ,θ):= 1(1+κ) c con 2 ( ρ 2 +2sρcosθ+ s 2 ) 1κ c con 2 ( ρ 2 +2sρcosθ+ s 2 ) .
γ ¯ 16 Nλ m,n [ ( a n m ) 2 + ( b n m ) 2 ] .
{ f ρ cosθ 1 ρ f θ sinθ, f ρ sinθ+ 1 ρ f θ cosθ,1},
z=[ 1 ρ max n a n 1 Q n 1 (0)] x+[ 1 ρ max n b n 1 Q n 1 (0) ]y+[ 1 2 c+ 1 ρ max 2 n a n 0 Q n 0 (0) ]( x 2 + y 2 ) +[ 1 ρ max 2 n a n 2 Q n 2 (0)] ( x 2 y 2 )+2[ 1 ρ max 2 n b n 2 Q n 2 (0)] xy+O(3).
P n m (x):= (1) n (2n)!! 2(2n1)!! P n ( 3 2 ,m 3 2 ) (2x1),
P n+1 m (x)=[ A n m + B n m x] P n m (x) C n m P n1 m (x),
A n m :=(2n1)(m+2n2)[ 4n(m+n2)+(m3)(2m1) ]/ D n m ,
B n m :=2(2n1)(m+2n3)(m+2n2)(m+2n1)/ D n m ,
C n m :=n(2n3)(m+2n1)(2m+2n3)/ D n m ,
D n m :=(4 n 2 1)(m+n2)(m+2n3).
P n m (x):={ 1x/2,m=1andn=1, (1) n (2n)!! 2(2n1)!! P n ( 3 2 ,m 3 2 ) (2x1),otherwise.
P 0 m (x)= 1 2 and P 1 m (x)={ 1x/2,m=1, m 1 2 (m1)x,m>1.
P 2 1 (x)=[3x(128x)]/6and P 3 1 (x)={ 5x[60x(12064x)] }/10.
x [ P n (a,b) (2x1)]=(n+a+b+1) P n1 (a+1,b+1) (2x1),
P n ( 3 2 ,m 3 2 ) (z)= 1 m+2n2 [ (m+n2) P n ( 1 2 ,m 3 2 ) (z)(m+n 3 2 ) P n1 ( 1 2 ,m 3 2 ) (z) ],
0 1 x m 3 2 [ P n ( 1 2 ,m 3 2 ) (2x1) ] 2 dx 1x =π (2n1)!!(2m+2n3)!! 2 m+2n1 (m+2n1)n!(m+n2)! ,
P n ( 1 2 ,m 3 2 ) (z)= 1 m+2n1 [ (m+n1) P n ( 1 2 ,m 1 2 ) (z)+(n 1 2 ) P n1 ( 1 2 ,m 1 2 ) (z) ],
x P n ( 1 2 ,m 1 2 ) (2x1)={ (n+1)(m+n) (m+2n+1)(m+2n) P n+1 ( 1 2 ,m 1 2 ) (2x1) + [4mn+m(2m1)+4 n 2 1] 2(m+2n+1)(m+2n1) P n ( 1 2 ,m 1 2 ) (2x1) + (2n1)(2m+2n1) 4(m+2n)(m+2n1) P n1 ( 1 2 ,m 1 2 ) (2x1) }.
F n m ={ m 2 (2m3)!! 2 m+1 (m1)! ,n=0, 4 (n1) 2 n 2 +1 8 (2n1) 2 + 11 32 δ n1 ,n>0andm=1, 2nχ(35m+4nχ)+ m 2 (3m+4nχ) (m+2n3)(m+2n2)(m+2n1)(2n1) γ n m ,n>0andm>1,
γ n m := n!(2m+2n3)!! 2 m+1 (m+n3)!(2n1)!! .
G n m ={ (2m1)!! 2 m+1 (m1)! ,n=0, (2 n 2 1)( n 2 1) 8(4 n 2 1) 1 24 δ n1 ,n>0andm=1, [2n(m+n1)m](n+1)(2m+2n1) (m+2n2)(m+2n1)(m+2n)(2n+1) γ n m ,n>0andm>1.
γ 1 m+1 = 2m+1 2(m1) γ 1 m and γ n+1 m = (n+1)(2m+2n1) (m+n2)(2n+1) γ n m .
P n m (x)={ f 0 m Q 0 m (x),n=0, f n m Q n m (x)+ g n1 m Q n1 m (x),n>0,
g n1 m = G n1 m / f n1 m ,and f n m = F n m g n1 m g n1 m .
[ F 0 1 F 1 1 F 2 1 F 3 1 F 0 2 F 1 2 F 2 2 F 3 2 F 0 3 F 1 3 F 2 3 F 3 3 F 0 4 F 1 4 F 2 4 F 3 4 ]=[ 1 4 15 32 17 72 29 40 1 2 7 8 35 36 67 40 27 32 35 16 35 16 243 80 5 4 511 128 23 6 12287 2560 ],[ G 0 1 G 1 1 G 2 1 G 0 1 G 1 2 G 2 2 G 0 1 G 1 3 G 2 3 G 0 1 G 1 4 G 2 4 ]=[ 1 4 1 24 7 40 3 8 5 48 7 16 15 32 7 32 117 160 35 64 21 64 33 32 ],
[ f 0 1 f 0 2 f 0 3 f 0 4 f 1 1 f 1 2 f 1 3 f 1 4 f 2 1 f 2 2 f 2 3 f 2 4 f 3 1 f 3 2 f 3 3 f 3 4 ]=[ 1 2 1 2 3 4 3 2 5 2 1 4 7 2 1 4 19 2 1 4 185 6 3 427 32 1 6 115 14 1 2 145 38 1 4 12803 370 1 2 2785 183 1 5 3397 230 1 4 6841 290 9 4 14113 25606 1 16 1289057 1114 ],
[ g 0 1 g 0 2 g 0 3 g 0 4 g 1 1 g 1 2 g 1 3 g 1 4 g 2 1 g 2 2 g 2 3 g 2 4 ]=[ 1 2 3 4 2 5 4 6 7 5 32 1 3 14 5 6 38 7 4 3 370 1 2 7 61 21 10 7 230 7 4 19 290 117 4 37 128030 33 16 183 2785 ].
Q n m (x)=[ P n m (x) g n1 m Q n1 m (x)]/ f n m .
P n m (0)={ 1,m=1andn=1, (2m+2n3)!! 2(2m3)!!(2n1)!! ,otherwise.
f(ρ,θ)= c ρ 2 1+ 1 c 2 ρ 2 + 1 1 c 2 ρ 2 { u 2 (1 u 2 ) n=0 N a n 0 Q n 0 ( u 2 ) + m=1 M u m [cos(mθ) n=0 N a n m Q n m ( u 2 ) +sin(mθ) n=0 N b n m Q n m ( u 2 ) ] }.
S(x):= n=0 N c n Q n m (x) ,
S(x) n=0 N d n P n m (x) ,
d n =( c n g n m d n+1 )/ f n m .
c n = f n m d n + g n m d n+1
α n = d n +( A n m + B n m x) α n+1 C n+1 m α n+2 .
A 0 1 =2, B 0 1 =1, A 1 1 =4/3, B 1 1 =8/3, C 1 1 =11/3, C 2 1 =0,
A 0 m =2m1, B 0 m =2(1m),whenm>1,
S(x)={ 1 2 α 0 2 5 α 3 ,m=1andN>2, 1 2 α 0 otherwise.
S (j) (x)={ 1 2 α 0 (j) 2 5 α 3 (j) ,m=1andN>2, 1 2 α 0 (j) otherwise,
α n (j) =j B n m α n+1 (j1) +( A n m + B n m x) α n+1 (j) C n+1 m α n+2 (j) ,whenj>0.
[ A 0 1 A 0 2 A 0 3 A 0 4 A 0 5 A 0 6 A 1 1 A 1 2 A 1 3 A 1 4 A 1 5 A 1 6 A 2 1 A 2 2 A 2 3 A 2 4 A 2 5 A 2 6 A 3 1 A 3 2 A 3 3 A 3 4 A 3 5 A 3 6 A 4 1 A 4 2 A 4 3 A 4 4 A 4 5 A 4 6 ]=[ 2 3 5 7 9 11 4 3 2 3 2 76 27 85 24 106 25 9 5 26 15 2 117 50 203 75 108 35 55 28 66 35 2 536 245 135 56 130 49 161 81 122 63 2 515 243 143 63 161 66 ],
[ B 0 1 B 0 2 B 0 3 B 0 4 B 0 5 B 0 6 B 1 1 B 1 2 B 1 3 B 1 4 B 1 5 B 1 6 B 2 1 B 2 2 B 2 3 B 2 4 B 2 5 B 2 6 B 3 1 B 3 2 B 3 3 B 3 4 B 3 5 B 3 6 B 4 1 B 4 2 B 4 3 B 4 4 B 4 5 B 4 6 ]=[ 1 2 4 6 8 10 8 3 4 4 40 9 5 28 5 24 5 4 4 21 5 112 25 24 5 30 7 4 4 144 35 30 7 220 49 112 27 4 4 110 27 88 21 13 3 ],
[ C 0 1 C 0 2 C 0 3 C 0 4 C 0 5 C 0 6 C 1 1 C 1 2 C 1 3 C 1 4 C 1 5 C 1 6 C 2 1 C 2 2 C 2 3 C 2 4 C 2 5 C 2 6 C 3 1 C 3 2 C 3 3 C 3 4 C 3 5 C 3 6 C 4 1 C 4 2 C 4 3 C 4 4 C 4 5 C 4 6 ]=[ * * * * * * 11 3 3 5 3 35 27 9 8 77 75 0 5 9 7 15 21 50 88 225 13 35 27 28 21 25 27 35 891 1225 39 56 33 49 80 81 45 49 55 63 1430 1701 40 49 1105 1386 ].

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