Abstract

Recent developments in technologies to correct aberrations in the eye have fostered extensive research in wave-front sensing of the eye, resulting in many reports of Zernike expansions of wave-front errors of the eye. For different reports of Zernike expansions, to be compared, the same pupil diameter is required. Since no standard pupil size has been established for reporting these results, a technique for converting Zernike expansion coefficients from one pupil size to another is needed. This investigation derives relationships between the Zernike expansion coefficients for two different pupil sizes.

© 2002 Optical Society of America

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References

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  1. J. Liang, W. Grimm, S. Goelz, J. F. Bille, “Objective measurement of the wave aberrations of the human eye us-ing a Hartmann–Shack wave-front sensor,” J. Opt. Soc. Am. A 7, 1949–1957 (1994).
    [CrossRef]
  2. P. Mierdel, M. Kaemmerer, M. Mrochen, H. E. Krinke, T. Seiler, “Ocular optical aberrometer for clinical use,” J. Biomed. Opt. 6, 200–204 (2001).
    [CrossRef] [PubMed]
  3. J. C. He, S. Marcos, R. H. Webb, S. A. Burns, “Measurement of the wave-front aberration of the eye by a fast psychophysical procedure,” J. Opt. Soc. Am. A 15, 2449–2456 (1998).
    [CrossRef]
  4. J. Porter, A. Guirao, I. G. Cox, D. R. Williams, “Monochromatic aberrations of the human eye in a large population,” J. Opt. Soc. Am. A 18, 1793–1803 (2001).
    [CrossRef]
  5. A. Bradley, X. Hong, L. N. Thibos, X. Cheng, D. T. Miller, “The statistics of monochromatic aberrations from 200 healthy young eyes,” Invest. Ophthalmol. Visual Sci. 42, S862 (2001).
  6. J. Y. Wang, D. E. Silva, “Wave-front interpretation with Zernike polynomials,” Appl. Opt. 19, 1510–1518 (1980).
    [CrossRef] [PubMed]
  7. L. N. Thibos, R. A. Applegate, H. C. Howland, D. R. Williams, P. Artal, R. Navarro, M. C. Campbell, J. E. Greivenkamp, J. T. Schwiegerling, S. A. Burns, D. A. Atchison, G. Smith, E. J. Sarver, “A VSIA-sponsored effort to develop methods and standards for the comparison of the wave-front aberration structure of the eye between devices and laboratories,” in Vision Sciences and Its Applications, Vol. 1 of 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1999), pp. 236–239.
  8. J. Straub, J. Schwiegerling, A. Gupta, “Design of a compact Shack–Hartmann aberrometer for real-time measurement of aberrations in human eyes,” in Vision Science and Its Applications, Vol. 53 of Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2001), pp. 110–113.
  9. K. A. Goldberg, K. Geary, “Wave-front measurement errors from restricted concentric subdomains,” J. Opt. Soc. Am. A 18, 2146–2152 (2001).
    [CrossRef]

2001 (4)

P. Mierdel, M. Kaemmerer, M. Mrochen, H. E. Krinke, T. Seiler, “Ocular optical aberrometer for clinical use,” J. Biomed. Opt. 6, 200–204 (2001).
[CrossRef] [PubMed]

A. Bradley, X. Hong, L. N. Thibos, X. Cheng, D. T. Miller, “The statistics of monochromatic aberrations from 200 healthy young eyes,” Invest. Ophthalmol. Visual Sci. 42, S862 (2001).

J. Porter, A. Guirao, I. G. Cox, D. R. Williams, “Monochromatic aberrations of the human eye in a large population,” J. Opt. Soc. Am. A 18, 1793–1803 (2001).
[CrossRef]

K. A. Goldberg, K. Geary, “Wave-front measurement errors from restricted concentric subdomains,” J. Opt. Soc. Am. A 18, 2146–2152 (2001).
[CrossRef]

1998 (1)

1994 (1)

J. Liang, W. Grimm, S. Goelz, J. F. Bille, “Objective measurement of the wave aberrations of the human eye us-ing a Hartmann–Shack wave-front sensor,” J. Opt. Soc. Am. A 7, 1949–1957 (1994).
[CrossRef]

1980 (1)

Applegate, R. A.

L. N. Thibos, R. A. Applegate, H. C. Howland, D. R. Williams, P. Artal, R. Navarro, M. C. Campbell, J. E. Greivenkamp, J. T. Schwiegerling, S. A. Burns, D. A. Atchison, G. Smith, E. J. Sarver, “A VSIA-sponsored effort to develop methods and standards for the comparison of the wave-front aberration structure of the eye between devices and laboratories,” in Vision Sciences and Its Applications, Vol. 1 of 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1999), pp. 236–239.

Artal, P.

L. N. Thibos, R. A. Applegate, H. C. Howland, D. R. Williams, P. Artal, R. Navarro, M. C. Campbell, J. E. Greivenkamp, J. T. Schwiegerling, S. A. Burns, D. A. Atchison, G. Smith, E. J. Sarver, “A VSIA-sponsored effort to develop methods and standards for the comparison of the wave-front aberration structure of the eye between devices and laboratories,” in Vision Sciences and Its Applications, Vol. 1 of 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1999), pp. 236–239.

Atchison, D. A.

L. N. Thibos, R. A. Applegate, H. C. Howland, D. R. Williams, P. Artal, R. Navarro, M. C. Campbell, J. E. Greivenkamp, J. T. Schwiegerling, S. A. Burns, D. A. Atchison, G. Smith, E. J. Sarver, “A VSIA-sponsored effort to develop methods and standards for the comparison of the wave-front aberration structure of the eye between devices and laboratories,” in Vision Sciences and Its Applications, Vol. 1 of 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1999), pp. 236–239.

Bille, J. F.

J. Liang, W. Grimm, S. Goelz, J. F. Bille, “Objective measurement of the wave aberrations of the human eye us-ing a Hartmann–Shack wave-front sensor,” J. Opt. Soc. Am. A 7, 1949–1957 (1994).
[CrossRef]

Bradley, A.

A. Bradley, X. Hong, L. N. Thibos, X. Cheng, D. T. Miller, “The statistics of monochromatic aberrations from 200 healthy young eyes,” Invest. Ophthalmol. Visual Sci. 42, S862 (2001).

Burns, S. A.

J. C. He, S. Marcos, R. H. Webb, S. A. Burns, “Measurement of the wave-front aberration of the eye by a fast psychophysical procedure,” J. Opt. Soc. Am. A 15, 2449–2456 (1998).
[CrossRef]

L. N. Thibos, R. A. Applegate, H. C. Howland, D. R. Williams, P. Artal, R. Navarro, M. C. Campbell, J. E. Greivenkamp, J. T. Schwiegerling, S. A. Burns, D. A. Atchison, G. Smith, E. J. Sarver, “A VSIA-sponsored effort to develop methods and standards for the comparison of the wave-front aberration structure of the eye between devices and laboratories,” in Vision Sciences and Its Applications, Vol. 1 of 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1999), pp. 236–239.

Campbell, M. C.

L. N. Thibos, R. A. Applegate, H. C. Howland, D. R. Williams, P. Artal, R. Navarro, M. C. Campbell, J. E. Greivenkamp, J. T. Schwiegerling, S. A. Burns, D. A. Atchison, G. Smith, E. J. Sarver, “A VSIA-sponsored effort to develop methods and standards for the comparison of the wave-front aberration structure of the eye between devices and laboratories,” in Vision Sciences and Its Applications, Vol. 1 of 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1999), pp. 236–239.

Cheng, X.

A. Bradley, X. Hong, L. N. Thibos, X. Cheng, D. T. Miller, “The statistics of monochromatic aberrations from 200 healthy young eyes,” Invest. Ophthalmol. Visual Sci. 42, S862 (2001).

Cox, I. G.

Geary, K.

Goelz, S.

J. Liang, W. Grimm, S. Goelz, J. F. Bille, “Objective measurement of the wave aberrations of the human eye us-ing a Hartmann–Shack wave-front sensor,” J. Opt. Soc. Am. A 7, 1949–1957 (1994).
[CrossRef]

Goldberg, K. A.

Greivenkamp, J. E.

L. N. Thibos, R. A. Applegate, H. C. Howland, D. R. Williams, P. Artal, R. Navarro, M. C. Campbell, J. E. Greivenkamp, J. T. Schwiegerling, S. A. Burns, D. A. Atchison, G. Smith, E. J. Sarver, “A VSIA-sponsored effort to develop methods and standards for the comparison of the wave-front aberration structure of the eye between devices and laboratories,” in Vision Sciences and Its Applications, Vol. 1 of 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1999), pp. 236–239.

Grimm, W.

J. Liang, W. Grimm, S. Goelz, J. F. Bille, “Objective measurement of the wave aberrations of the human eye us-ing a Hartmann–Shack wave-front sensor,” J. Opt. Soc. Am. A 7, 1949–1957 (1994).
[CrossRef]

Guirao, A.

Gupta, A.

J. Straub, J. Schwiegerling, A. Gupta, “Design of a compact Shack–Hartmann aberrometer for real-time measurement of aberrations in human eyes,” in Vision Science and Its Applications, Vol. 53 of Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2001), pp. 110–113.

He, J. C.

Hong, X.

A. Bradley, X. Hong, L. N. Thibos, X. Cheng, D. T. Miller, “The statistics of monochromatic aberrations from 200 healthy young eyes,” Invest. Ophthalmol. Visual Sci. 42, S862 (2001).

Howland, H. C.

L. N. Thibos, R. A. Applegate, H. C. Howland, D. R. Williams, P. Artal, R. Navarro, M. C. Campbell, J. E. Greivenkamp, J. T. Schwiegerling, S. A. Burns, D. A. Atchison, G. Smith, E. J. Sarver, “A VSIA-sponsored effort to develop methods and standards for the comparison of the wave-front aberration structure of the eye between devices and laboratories,” in Vision Sciences and Its Applications, Vol. 1 of 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1999), pp. 236–239.

Kaemmerer, M.

P. Mierdel, M. Kaemmerer, M. Mrochen, H. E. Krinke, T. Seiler, “Ocular optical aberrometer for clinical use,” J. Biomed. Opt. 6, 200–204 (2001).
[CrossRef] [PubMed]

Krinke, H. E.

P. Mierdel, M. Kaemmerer, M. Mrochen, H. E. Krinke, T. Seiler, “Ocular optical aberrometer for clinical use,” J. Biomed. Opt. 6, 200–204 (2001).
[CrossRef] [PubMed]

Liang, J.

J. Liang, W. Grimm, S. Goelz, J. F. Bille, “Objective measurement of the wave aberrations of the human eye us-ing a Hartmann–Shack wave-front sensor,” J. Opt. Soc. Am. A 7, 1949–1957 (1994).
[CrossRef]

Marcos, S.

Mierdel, P.

P. Mierdel, M. Kaemmerer, M. Mrochen, H. E. Krinke, T. Seiler, “Ocular optical aberrometer for clinical use,” J. Biomed. Opt. 6, 200–204 (2001).
[CrossRef] [PubMed]

Miller, D. T.

A. Bradley, X. Hong, L. N. Thibos, X. Cheng, D. T. Miller, “The statistics of monochromatic aberrations from 200 healthy young eyes,” Invest. Ophthalmol. Visual Sci. 42, S862 (2001).

Mrochen, M.

P. Mierdel, M. Kaemmerer, M. Mrochen, H. E. Krinke, T. Seiler, “Ocular optical aberrometer for clinical use,” J. Biomed. Opt. 6, 200–204 (2001).
[CrossRef] [PubMed]

Navarro, R.

L. N. Thibos, R. A. Applegate, H. C. Howland, D. R. Williams, P. Artal, R. Navarro, M. C. Campbell, J. E. Greivenkamp, J. T. Schwiegerling, S. A. Burns, D. A. Atchison, G. Smith, E. J. Sarver, “A VSIA-sponsored effort to develop methods and standards for the comparison of the wave-front aberration structure of the eye between devices and laboratories,” in Vision Sciences and Its Applications, Vol. 1 of 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1999), pp. 236–239.

Porter, J.

Sarver, E. J.

L. N. Thibos, R. A. Applegate, H. C. Howland, D. R. Williams, P. Artal, R. Navarro, M. C. Campbell, J. E. Greivenkamp, J. T. Schwiegerling, S. A. Burns, D. A. Atchison, G. Smith, E. J. Sarver, “A VSIA-sponsored effort to develop methods and standards for the comparison of the wave-front aberration structure of the eye between devices and laboratories,” in Vision Sciences and Its Applications, Vol. 1 of 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1999), pp. 236–239.

Schwiegerling, J.

J. Straub, J. Schwiegerling, A. Gupta, “Design of a compact Shack–Hartmann aberrometer for real-time measurement of aberrations in human eyes,” in Vision Science and Its Applications, Vol. 53 of Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2001), pp. 110–113.

Schwiegerling, J. T.

L. N. Thibos, R. A. Applegate, H. C. Howland, D. R. Williams, P. Artal, R. Navarro, M. C. Campbell, J. E. Greivenkamp, J. T. Schwiegerling, S. A. Burns, D. A. Atchison, G. Smith, E. J. Sarver, “A VSIA-sponsored effort to develop methods and standards for the comparison of the wave-front aberration structure of the eye between devices and laboratories,” in Vision Sciences and Its Applications, Vol. 1 of 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1999), pp. 236–239.

Seiler, T.

P. Mierdel, M. Kaemmerer, M. Mrochen, H. E. Krinke, T. Seiler, “Ocular optical aberrometer for clinical use,” J. Biomed. Opt. 6, 200–204 (2001).
[CrossRef] [PubMed]

Silva, D. E.

Smith, G.

L. N. Thibos, R. A. Applegate, H. C. Howland, D. R. Williams, P. Artal, R. Navarro, M. C. Campbell, J. E. Greivenkamp, J. T. Schwiegerling, S. A. Burns, D. A. Atchison, G. Smith, E. J. Sarver, “A VSIA-sponsored effort to develop methods and standards for the comparison of the wave-front aberration structure of the eye between devices and laboratories,” in Vision Sciences and Its Applications, Vol. 1 of 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1999), pp. 236–239.

Straub, J.

J. Straub, J. Schwiegerling, A. Gupta, “Design of a compact Shack–Hartmann aberrometer for real-time measurement of aberrations in human eyes,” in Vision Science and Its Applications, Vol. 53 of Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2001), pp. 110–113.

Thibos, L. N.

A. Bradley, X. Hong, L. N. Thibos, X. Cheng, D. T. Miller, “The statistics of monochromatic aberrations from 200 healthy young eyes,” Invest. Ophthalmol. Visual Sci. 42, S862 (2001).

L. N. Thibos, R. A. Applegate, H. C. Howland, D. R. Williams, P. Artal, R. Navarro, M. C. Campbell, J. E. Greivenkamp, J. T. Schwiegerling, S. A. Burns, D. A. Atchison, G. Smith, E. J. Sarver, “A VSIA-sponsored effort to develop methods and standards for the comparison of the wave-front aberration structure of the eye between devices and laboratories,” in Vision Sciences and Its Applications, Vol. 1 of 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1999), pp. 236–239.

Wang, J. Y.

Webb, R. H.

Williams, D. R.

J. Porter, A. Guirao, I. G. Cox, D. R. Williams, “Monochromatic aberrations of the human eye in a large population,” J. Opt. Soc. Am. A 18, 1793–1803 (2001).
[CrossRef]

L. N. Thibos, R. A. Applegate, H. C. Howland, D. R. Williams, P. Artal, R. Navarro, M. C. Campbell, J. E. Greivenkamp, J. T. Schwiegerling, S. A. Burns, D. A. Atchison, G. Smith, E. J. Sarver, “A VSIA-sponsored effort to develop methods and standards for the comparison of the wave-front aberration structure of the eye between devices and laboratories,” in Vision Sciences and Its Applications, Vol. 1 of 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1999), pp. 236–239.

Appl. Opt. (1)

Invest. Ophthalmol. Visual Sci. (1)

A. Bradley, X. Hong, L. N. Thibos, X. Cheng, D. T. Miller, “The statistics of monochromatic aberrations from 200 healthy young eyes,” Invest. Ophthalmol. Visual Sci. 42, S862 (2001).

J. Biomed. Opt. (1)

P. Mierdel, M. Kaemmerer, M. Mrochen, H. E. Krinke, T. Seiler, “Ocular optical aberrometer for clinical use,” J. Biomed. Opt. 6, 200–204 (2001).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (4)

Other (2)

L. N. Thibos, R. A. Applegate, H. C. Howland, D. R. Williams, P. Artal, R. Navarro, M. C. Campbell, J. E. Greivenkamp, J. T. Schwiegerling, S. A. Burns, D. A. Atchison, G. Smith, E. J. Sarver, “A VSIA-sponsored effort to develop methods and standards for the comparison of the wave-front aberration structure of the eye between devices and laboratories,” in Vision Sciences and Its Applications, Vol. 1 of 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1999), pp. 236–239.

J. Straub, J. Schwiegerling, A. Gupta, “Design of a compact Shack–Hartmann aberrometer for real-time measurement of aberrations in human eyes,” in Vision Science and Its Applications, Vol. 53 of Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2001), pp. 110–113.

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Figures (5)

Fig. 1
Fig. 1

Conventional right-handed coordinate system for representing wave-front error in the eye.

Fig. 2
Fig. 2

(a) Superposition of the subject’s wave-front-error functions based on the original coefficients anm and the new coefficients bnm. (b) Difference between the two reconstructions of the wave-front error over the region r3.000 mm.

Fig. 3
Fig. 3

Difference between the wave front generated by the resampling method and the original wave front along the horizontal meridian.

Fig. 4
Fig. 4

Difference between the wave front generated by the fixed-pupil method and the original wave front along the horizontal meridian.

Fig. 5
Fig. 5

Spherical aberration coefficient b40 as a function of pupil size r2.

Tables (2)

Tables Icon

Table 1 Coefficient-Conversion Relationships for Zernike Polynomial Expansion up to 36 Terms (Seventh Radial Order)

Tables Icon

Table 2 Original Expansion Coefficients anm for a Zernike Fit with r1=3.665 mm and the New Expansion Coefficients bnm for a Zernike Fit with r2=3.000 mm.

Equations (55)

Equations on this page are rendered with MathJax. Learn more.

Znm(ρ, θ)=NnmRn|m|(ρ)cosmθform0-NnmRn|m|(ρ)sinmθform<0
Rn|m|(ρ)
=s=0(n-|m|)/2(-1)s(n-s)!s!0.5(n+|m|)-s!0.5(n-|m|)-s!
×ρn-2s.
Nnm=2(n+1)1+δm0,
W(r, θ)=n,man,mZnm(r/rmax, θ),
W(r, θ)=n,man,mZnm(r/r1, θ)
W(r, θ)=n,mbn,mZnm(r/r2, θ),
a00+a2032r2r12-1+a4056r4r14-6r2r12+1.
a00-a203+a405+a20r22r1232r2r22
-a40r22r1256r2r22+a40r24r1456r4r24.
b40=a40r24r14.
a00-a203+a405-b405+a20r22r1232r2r22
-a40r22r1256r2r22+b4056r2r22
+b4056r4r24-b4056r2r22+b405.
a00-a203+a405-b405+r22r12[a20-a4015
+b4015]32r2r22+b40Z40(r/r2, θ).
b20=r22r12[a20-a4015+b4015].
a00-a203+a405-b405+b203
+b2032r2r22-b203+b40Z40(r/r2, θ).
b00=a00-a203+b203+a405-b405,
b00+b2Z20(r/r2, θ)+b40Z40(r/r2, θ).
b40=r243.26640.00099385-0.00046915 * 351-r223.2662.
a|m|,m2(|m|+1)1+δm0 R|m||m|rr1
+a|m|+2,m2(|m|+2+1)1+δm0 R|m|+2|m|rr1
+a|m|+4,m2(|m|+4+1)1+δm0 R|m|+4|m|rr1
++aN,m2(N+1)1+δm0 RN|m|rr1
×cos mθ,m0;×-sin mθ,m<0;
b|m|,m2(|m|+1)1+δm0 R|m||m|rr2+b|m|+2,m
×2(|m|+2+1)1+δm0 R|m|+2|m|rr2+b|m|+4,m
×2(|m|+4+1)1+δm0 R|m|+4|m|rr2++bN,m
×2(N+1)1+δm0 RN|m|rr2
×cos mθ,m0;×-sin mθ,m<0.
a|m|,m2(|m|+1)1+δm0s=00(-1)s(|m|-s)!s!(|m|-s)!(-s)!rr1|m|-2s
+a|m|+2,m2(|m|+2+1)1+δm0
×s=01(-1)s(|m|+2-s)!s!(|m|+1-s)!(1-s)!rr1|m|+2-2s
+a|m|+4,m2(|m|+4+1)1+δm0
×s=02(-1)s(|m|+4-s)!s!(|m|+2-s)!(2-s)!rr1|m|+4-2s+
+aN,m2(N+1)1+δm0s=0(N-|m|)/2[(-1)s(N-s)!]
s!N+|m|2-s!N-|m|2-s!rr1N-2s.
j=0(N-|m|)/2(-1)ja|m|+2j,m2(|m|+2j+1)1+δm0(|m|+j)!j!|m|!0!r|m|r1|m|r2|m|r2|m|2(|m|+1)1+δm0|m|!|m|!0!1+δm02(|m|+1)|m|!0!|m|!
+j=0(N-|m|-2)/2(-1)ja|m|+2+2j,m2(|m|+2+2j+1)1+δm0(|m|+2+j)!j!(|m|+1)!1!r|m|+2r1|m|+2r2|m|+2r2|m|+2
×2(|m|+3)1+δm0(|m|+2)!(|m|+1)!1!1+δm02(|m|+3)(|m|+1)!1!(|m|+2)!
+j=0(N-|m|-4)/2(-1)ja|m|+4+2j,m2(|m|+4+2j+1)1+δm0(|m|+4+j)!j!(|m|+2)!2!r|m|+4r1|m|+4r2|m|+4r2|m|+4
×2(|m|+5)1+δm0(|m|+4)!(|m|+2)!2!1+δm02(|m|+5)(|m|+2)!2!(|m|+4)!++j=00(-1)jaN+2j,m
×2(N+2j+1)1+δm0(N+j)!j![(N+|m|)/2]![(N-|m|)/2]!rNr1Nr2Nr2N
×2(N+1)1+δm0 N!N+|m|2!N-|m|2! 1+δm02(N+1) N+|m|2!N-|m|2N!.
r2|m|r1|m|j=0(N-|m|)/2(-1)ja|m|+2j,m|m|+2j+1|m|+1(|m|+j)!j!|m|!|m|!|m|!0!2(|m|+1)1+δm0r|m|r2|m|
+r2|m|+2r1|m|+2j=0(N-|m|-2)/2(-1)ja|m|+2+2j,m|m|+2j+3|m|+3(|m|+2+j)!j!(|m|+2)!(|m|+2)!(|m|+1)!1!2(|m|+3)1+δm0r|m|+2r2|m|+2
+r2|m|+4r1|m|+4j=0(N-|m|-4)/2(-1)ja|m|+4+2j,m|m|+2j+5|m|+5(|m|+4+j)!j!(|m|+4)!(|m|+4)!(|m|+2)!2!2(|m|+5)1+δm0r|m|+4r2|m|+4+
+r2Nr1Nj=00(-1)jaN+2j,mN+2j+1N+1(N+j)!j!N! N!N+|m|2!N-|m|2! 2(N+1)1+δm0rNr2N.
bNm=r2Nr1NaNm.
bNm2(N+1)1+δm0 s=1(N-|m|)/2(-1)s(N-s)!
s!N+|m|2-s!N-|m|2-s! rr2N-2s
b|m|+2i,m=r2|m|+2ir1|m|+2ij=0(N-|m|-2i)/2(-1)ja|m|+2(i+j),m×|m|+2(i+j)+1|m|+2i+1(|m|+2i+j)!j!(|m|+2i)!-k=|m|+2(i+1)step 2Nbkmk+1|m|+2i+1×(-1)(k-|m|-2i)/2[(k+|m|+2i)/2]![(k-|m|-2i)/2]!(|m|+2i)!×r1|m|-2ir2|m|+2i,

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