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J. Eur. Opt. Society-Rapid Publ. 22, 31( 2026)
of surfaces using their polynomial coefficients. Theoretically, this significantly improves computational speed, reduces memory requirements, and increases application flexibility without compromising accuracy. Optical applications can benefit from these new insights.
This work provides a solid starting point for further theoretical and application-specific studies, such as a comprehensive evaluation of computational complexity, a dedicated algorithmic accuracy / stability investigation, and application-oriented algorithm design, Zernike analysis with non-circular pupil, fast higher-order derivatives of a surface described by Zernike polynomials, etc. It is also valuable for discovering similar cases of block-wise recurrence in other orthogonal polynomials constructed using the Gram-Schmidt process.
Funding This research received no external funding.
Conflicts of interest
The authors declare no conflicts of interest in regards to this article.
Data availability statement
The underlying data can be provided by the authors upon reasonable request.
References
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Appendix A: Mathematical proofs
A. 1 Pascal triangle supported block-wise recurrence in Zernike polynomials
The Zernike radial polynomial has its binomial form( equation( 8)). Defining n = 2j, m = 2l forevenordersandn = 2j + 1, m = 2l + 1 for odd orders, and defining two intermediate variables
J k = j � k and J 2k = 2J k, weobtain
8
R 2l 2j ¼ Pj�l ð�1Þ k J 2k þ k J 2k >< ðq 2 Þ J k k¼0 J 2k J k þ l
R 2lþ1 2jþ1; ¼ q Pj�l ð�1Þ k J 2k þ k þ J 2k þ k
>: J 2k þ 1 k¼0
J 2k
J 2k
J k þ l
þ
Further defining w e k ¼ J
2k þ k, w
J o k ¼ we 2k
t e lk ¼ J 2k
, andt J k þ l o lk ¼ te lk þ J 2k
J k þ l þ 1
8
R 2l 2j ¼ Pj�l ð�1Þ k w e k te lk ðq2 Þ J k
><
>: k¼0
R 2lþ1 2jþ1 ¼ q Pj�l k¼0 ð�1Þ k w o k to lk ðq2 Þ J k
J 2k J k þ l þ 1
k þ
, weobtain
ðq 2 Þ J k
: ðA: 1Þ
J 2k þ k
,
J 2k þ 1: ðA: 2Þ
In particular, we have w e k ¼ J
2k þ k 2j � k ¼, forwhich
J 2k k we obtain a vector by continuously increasing k by 1 from 0 to
k = j � l, provided w e fkg ¼ 2j 2j � 1 j þ l as a
0 1 j � l row vector, where 2j = n forevenordersandl j; wealsohave
w o k ¼ we k þ J 2k þ k
J 2k þ 1 h provided w o fkg ¼
¼
2j þ 1 0
2j � k k
þ 2j 1
2j � k k � 1
¼
2j þ 1 � k
, k i
, where j þ 1 þ l j � l
2j + 1 = n for odd orders and l j. Eitherw e fkg or wo fkg denotes a corresponding anti-diagonal vector in a left-aligned n-row Pascal triangle( e. g., Table 2 for n 6). For one even( or odd) order, w e fkg( or wo fkg
) depends on a single variable k and is independent of l or m.
Meanwhile, for an even order n = 2j, we have t e lk ¼
J 2k
J k þ l
¼
2ðj � kÞ ðj � kÞþl
, where t e lk j k¼0 ¼ 2j j þ l
subject to k = 0; continuously increasing k by 1 till k = j � l, we obtain
h a row vector t e lfkg ¼ 2j 2j � 2 2j � 4
j þ l j � 1 þ l j � 2 þ l i
2l with respect to a particular l, and obtain an anti-upper-triangular matrix for all the 0 l j,
2l as