J. Eur. Opt. Society-Rapid Publ. 22, 31( 2026) 319
2 2j
6 j
2j
j þ 1
2j
¼ j þ 2 2j
6
2j � 2
4 2j 2j
2j � 2
j � 1
2j � 2
j
2j � 2 j þ 1
2j � 2 2j � 2
� T n¼2j t e lk
2j � 4
j � 2
2j � 4
j � 1
2j � 4 j...
¼ t e flgfkg
2
1 2 2
0 0
3
7 5
ðjþ1Þðjþ1Þ
: ðA: 3Þ
Also for an odd order n = 2j + 1, we have
t o lk ¼ te lk þ J 2k
2ðj � kÞ 2ðj � kÞ ¼ þ ¼
J k þ l þ 1 ðj � kÞþl ðj � kÞþl þ 1
2ðj � kÞþ1, and an anti-upper-triangular matrix as follows: ððj � kÞþ1Þþl
� T n¼2jþ1 t o lk ¼ t o flgfkg
|
2
2j þ 1
|
2j � 1
|
2j � 3
|
|
j þ 1
2j þ 1 j þ 2
2j þ 1 j þ 3
2j þ 1
6
2j
4
2j þ 1
|
j
2j � 1
j þ 1
2j � 1 j þ 2
2j � 1
2j � 1
|
j � 1
2j � 3
j
2j � 3 j þ 1
.
..
|
2j þ 1 |
|
|
3
2 3 3
1 1
3
7 5
ðjþ1Þðjþ1Þ
: ðA: 4Þ
The results of mathematical derivation starting from binomial equivalence agree with the intuitive visual observation in the previous Section 2.1:
8 ð�1Þ k w e k wk n¼2j
ð�1Þ k w o k wk n¼2jþ1 >< t e lk j k¼0 tm¼2l n¼2j
t o lk j k¼0 tm¼2lþ1 n¼2jþ1
T n t o = e lk = 0 T n ¼ >: t fmg n
T n�2
0
; ðA: 5Þ
where T n t o = e lk = 0 denotes a variation of T n t o = e lk by filling the elements below the anti-diagonal elements with zeros. Furthermore, by adopting
8
><
>:
diag w fkg
2j
diag w fkg
2jþ1
2
¼
6 4 ð�1Þ 0 w e k¼0
2 ð�1Þ 0 w o k¼0
¼;
6 4 ð�1Þ 1 w e k¼1
ð�1Þ 1 w o k¼1
...
...
3
7 5 ð�1Þ j�l w e k¼j
3
7 5 ð�1Þ j�l w o k¼j ðjþ1Þðjþ1Þ
ðjþ1Þðjþ1Þ
ðA: 6Þ
;
Equation( A. 2) can be re-written as
8 ><
>:
R f2lg 2j
R f2lþ1g 2jþ1
¼ T 2j diag w fkg
2j Þ ðq 2 Þ j ðq 2 Þ j�1 ðq 2 T
Þ 1
¼ T 2jþ1 diag w fkg
2jþ1 ðq 2 Þ j ðq 2 Þ j�1 ðq 2 Þ 1
ðjþ1Þ1
T
ðjþ1Þ1
;
q ðA: 7Þ
where the radial coordinate-dependent vector ðq 2 Þ j ðq 2 Þ j�1 ðq 2 T
Þ 1 contains all possible ðq 2 Þ J k in equation( A. 7), " k 2 [ 0, j � l ], " l 2 [ 0, j ]. Essentially equation( A. 7) is equivalent to equation( 5), and equations( A. 3) and(( A. 4)) are equivalent to equation( 7), without an order limit(" n 0).
It is noticed that the close relationship between the radial polynomials and Pascal triangle can be straightforwardly obtained even without the intermediate variables J k and J 2k: The ordered
() n � k set of binomial factors k ¼ 0; 1;::: ðn � mÞ = 2 corresponds to an anti-diagonal vector of a left-aligned Pascal triangle,
k
where k increases while n � k decreases, each in steps of 1, as
shown in Table 2 for n 6; the binomial factor
n � 2k can be rewritten as
ðn � mÞ = 2 � k
n � 2k
, which is equivalent to
ðn � 2kÞ = 2 � m = 2
n � 2k
. If k varies as the column index and ðn � 2kÞ = 2 þ m = 2
l = m / 2forevenordersorl =( m � 1)/ 2 for odd orders varies as the row index, the ordered factor set
n � 2k ðn � 2kÞ = 2 þ l l ¼ 0; 1; j; k ¼ 0; 1;::: j � l forms a 2D
n � 2k matrix. For a certain k, the column vector ðn � 2kÞ = 2 þflg corresponds to the right half of the( n � 2k) th row in Pascal triangle( Table 1 for n 6). Nevertheless, using the intermediate variables J k and J 2k, additional block-wise relationships such as
w o k ¼ we k þ J
2k þ k and t J 2k þ 1 o lk ¼ te lk þ J 2k can be
J k þ l þ 1 obtained, which, although not as intuitive, are useful for further algorithm development. A. 2 Zernike radial polynomials in Cartesian: the second blockwise recurrence
Considering that q 2 = x 2 + y 2, its power value( q 2) j can be expressed as the multiplication of two vectors:
� q 2 j
¼ Xj q¼0 b q j x2ðj�qÞ y 2q ¼ b fqg j
X fqg j ðA: 8Þ where
b fqg j denotes a row vector containing all binomial factors of fb q j
¼ j jq ¼ 0; 1; 2;::: jg, corresponding to the jth row in an n- q row Pascal triangle, and X fqg j denotes a column vector of the sequentially listed jth basis functions of homogeneous bivariate
polynomials for( x 2, y 2), as x 2j y 0 x 2ðj�1Þ y 2 x 0 y 2j T
.