User:RDBury/Scratchpad

Extended quotient rule

From the generalized product rule, if h=fg then

${\displaystyle h'=fg'+f'g\,}$

${\displaystyle h''=fg''+2f'g'+f''g\,}$

${\displaystyle h'''=fg'''+3f'g''+3f''g'+f'''g\,}$

${\displaystyle \,\,\vdots }$

${\displaystyle h^{(n)}=fg^{(n)}+{n \choose 1}f'g^{(n-1)}+{n \choose 2}f''g^{(n-2)}+\dots +f^{(n)}g}$

Using Cramer's rule to solve for f⁽ⁿ⁾ produces the determinant formula ^[1]

${\displaystyle f^{(n)}=\left({\frac {h}{g}}\right)^{(n)}=}$

${\displaystyle g^{-(n+1)}{\begin{vmatrix}g&&&&&h\\g'&g&&&&h'\\g''&2g'&g&&&h''\\g'''&3g''&3g'&g&&h'''\\\vdots &&&&\ddots &\\g^{(n)}&{n \choose 1}g^{(n-1)}&{n \choose 2}g^{(n-2)}&{n \choose 3}g^{(n-3)}&\cdots &h^{(n)}\end{vmatrix}}}$

By applying this to find Taylor series coefficients in the cases h=x, g=e^x-1; h=e^x-1, g=e^x+1; h=sin x, g=cos x; h=x, g=sin x; and h=1, g=cos x; four different determinant expressions for the Bernoulli numbers and a determinant expression for the Euler numbers can be obtained.^[2]

Symmetric polynomials

The Schur polynomial

${\displaystyle s_{(d_{1},d_{2},\dots ,d_{n})}(x_{1},x_{2},\dots ,x_{n})}$

are defined as the quotients of the alternating polynomial

${\displaystyle {\begin{vmatrix}x_{1}^{d_{1}+n-1}&x_{2}^{d_{1}+n-1}&\cdots &x_{n}^{d_{1}+n-1}\\x_{1}^{d_{2}+n-2}&x_{2}^{d_{2}+n-2}&\cdots &x_{n}^{d_{2}+n-2}\\\vdots &\vdots &\ddots &\vdots \\x_{1}^{d_{n}}&x_{2}^{d_{n}}&\cdots &x_{n}^{d_{n}}\end{vmatrix}}}$

and the Vandermond determinant

${\displaystyle {\begin{vmatrix}x_{1}^{n-1}&x_{2}^{n-1}&\cdots &x_{n}^{n-1}\\x_{1}^{n-2}&x_{2}^{n-2}&\cdots &x_{n}^{n-2}\\\vdots &\vdots &\ddots &\vdots \\1&1&\cdots &1\end{vmatrix}}}$

This can, in turn, be expressed as a determinant involving the complete homogeneous symmetric polynomials as^[3]

${\displaystyle {\begin{vmatrix}h_{d_{1}+n-1}&h_{d_{1}+n-2}&\cdots &h_{d_{1}}\\h_{d_{2}+n-2}&h_{d_{2}+n-3}&\cdots &h_{d_{2}-1}\\\vdots &\vdots &\ddots &\vdots \\h_{d_{n}}&h_{d_{n}-1}&\cdots &h_{d_{n}-n+1}\\\end{vmatrix}}}$

Newton's identities A002135 Number of terms in a symmetric determinant (See Muir p. 112)