向下克倫肖遞推公式計算指標係數與服從遞推關係的函式的乘積之和。如果
 |
(1)
|
並且
 |
(2)
|
其中 
是已知的，則定義
 |  |  |
(3)
|
 |  |  |
(4)
|
對於 
並向後求解以獲得 
和 
。
 |
(5)
|
 |  |  |
(6)
|
 |  | ![c_0F_0(x)+[y_1-alpha(1,x)y_2-beta(2,x)y_3]F_1(x)+[y_2-alpha(2,x)y_3-beta(3,x)y_4]F_2(x)+[y_3-alpha(3,x)y_4-beta(4,x)y_5]F_3(x)+[y_4-alpha(4,x)y_5-beta(5,x)y_6]F_4(x)+...](/images/equations/ClenshawRecurrenceFormula/Inline16.svg) |
(7)
|
 |  | ![c_0F_0(x)+y_1F_1(x)+y_2[F_2(x)-alpha(1,x)F_1(x)]+y_3[F_3(x)-alpha(2,x)F_2(x)-F_1(x)beta(2,x)]+y_4[F_4(x)-alpha(3,x)F_3(x)-F_2(x)beta(3,x)]+...](/images/equations/ClenshawRecurrenceFormula/Inline19.svg) |
(8)
|
 |  | ![c_0F_0(x)+y_2[{alpha(1,x)F_1(x)+beta(1,x)F_0(x)}-alpha(1,x)F_1(x)]+y_1F_1(x)](/images/equations/ClenshawRecurrenceFormula/Inline22.svg) |
(9)
|
 |  |  |
(10)
|
向上克倫肖遞推公式為
 |
(11)
|
![y_k=1/(beta(k+1,x))[y_(k-2)-alpha(k,x)y_(k-1)-c_k]](/images/equations/ClenshawRecurrenceFormula/NumberedEquation5.svg) |
(12)
|
對於 
。
 |
(13)
|
克倫肖遞推公式
使用 探索
參考資料
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. "Recurrence Relations and Clenshaw's Recurrence Formula." §5.5 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 172-178, 1992.在 上被引用
克倫肖遞推公式引用為
Weisstein, Eric W. "Clenshaw Recurrence Formula." 來自 -- Wolfram 網路資源. https://mathworld.tw/ClenshawRecurrenceFormula.html