Formula for the application of a linear differential operator to the product of exponential and polynomial...












3












$begingroup$


In the context of linear differential equations, I've stumbled upon the following identity for an arbitrary pair of polynomials $P$ and $Q$ with real or complex coefficients:
$$
Pleft(frac{d}{dx}right)bigl(e^{xy}Q(x)bigr)
=sum_{n=0}^inftyfrac{P^{(n)}(y)e^{xy}Q^{(n)}(x)}{n!}
= Qleft(frac{d}{dy}right)bigl(e^{xy}P(y)bigr).
$$

This can be more or less easily checked by using Taylor expansions of $Pbigl(frac{d}{dx}bigr)$ at $y$ and of $Qbigl(frac{d}{dy}bigr)$ at $x$:
$$
Pleft(frac{d}{dx}right)
=sum_{n=0}^inftyfrac{P^{(n)}(y)}{n!}left(frac{d}{dx} - yright)^n,
quad
Qleft(frac{d}{dy}right)
=sum_{n=0}^inftyfrac{Q^{(n)}(x)}{n!}left(frac{d}{dy} - xright)^n.
$$



Is there any easy way to "see" that
$Pbigl(frac{d}{dx}bigr)bigl(e^{xy}Q(x)) = Qbigl(frac{d}{dy}bigr)bigl(e^{xy}P(y)bigr)$
without "getting hands dirty"?



Is this identity a part of some general theory?
It makes me think of Fourier analysis, but I do not know much about it.










share|cite|improve this question











$endgroup$

















    3












    $begingroup$


    In the context of linear differential equations, I've stumbled upon the following identity for an arbitrary pair of polynomials $P$ and $Q$ with real or complex coefficients:
    $$
    Pleft(frac{d}{dx}right)bigl(e^{xy}Q(x)bigr)
    =sum_{n=0}^inftyfrac{P^{(n)}(y)e^{xy}Q^{(n)}(x)}{n!}
    = Qleft(frac{d}{dy}right)bigl(e^{xy}P(y)bigr).
    $$

    This can be more or less easily checked by using Taylor expansions of $Pbigl(frac{d}{dx}bigr)$ at $y$ and of $Qbigl(frac{d}{dy}bigr)$ at $x$:
    $$
    Pleft(frac{d}{dx}right)
    =sum_{n=0}^inftyfrac{P^{(n)}(y)}{n!}left(frac{d}{dx} - yright)^n,
    quad
    Qleft(frac{d}{dy}right)
    =sum_{n=0}^inftyfrac{Q^{(n)}(x)}{n!}left(frac{d}{dy} - xright)^n.
    $$



    Is there any easy way to "see" that
    $Pbigl(frac{d}{dx}bigr)bigl(e^{xy}Q(x)) = Qbigl(frac{d}{dy}bigr)bigl(e^{xy}P(y)bigr)$
    without "getting hands dirty"?



    Is this identity a part of some general theory?
    It makes me think of Fourier analysis, but I do not know much about it.










    share|cite|improve this question











    $endgroup$















      3












      3








      3





      $begingroup$


      In the context of linear differential equations, I've stumbled upon the following identity for an arbitrary pair of polynomials $P$ and $Q$ with real or complex coefficients:
      $$
      Pleft(frac{d}{dx}right)bigl(e^{xy}Q(x)bigr)
      =sum_{n=0}^inftyfrac{P^{(n)}(y)e^{xy}Q^{(n)}(x)}{n!}
      = Qleft(frac{d}{dy}right)bigl(e^{xy}P(y)bigr).
      $$

      This can be more or less easily checked by using Taylor expansions of $Pbigl(frac{d}{dx}bigr)$ at $y$ and of $Qbigl(frac{d}{dy}bigr)$ at $x$:
      $$
      Pleft(frac{d}{dx}right)
      =sum_{n=0}^inftyfrac{P^{(n)}(y)}{n!}left(frac{d}{dx} - yright)^n,
      quad
      Qleft(frac{d}{dy}right)
      =sum_{n=0}^inftyfrac{Q^{(n)}(x)}{n!}left(frac{d}{dy} - xright)^n.
      $$



      Is there any easy way to "see" that
      $Pbigl(frac{d}{dx}bigr)bigl(e^{xy}Q(x)) = Qbigl(frac{d}{dy}bigr)bigl(e^{xy}P(y)bigr)$
      without "getting hands dirty"?



      Is this identity a part of some general theory?
      It makes me think of Fourier analysis, but I do not know much about it.










      share|cite|improve this question











      $endgroup$




      In the context of linear differential equations, I've stumbled upon the following identity for an arbitrary pair of polynomials $P$ and $Q$ with real or complex coefficients:
      $$
      Pleft(frac{d}{dx}right)bigl(e^{xy}Q(x)bigr)
      =sum_{n=0}^inftyfrac{P^{(n)}(y)e^{xy}Q^{(n)}(x)}{n!}
      = Qleft(frac{d}{dy}right)bigl(e^{xy}P(y)bigr).
      $$

      This can be more or less easily checked by using Taylor expansions of $Pbigl(frac{d}{dx}bigr)$ at $y$ and of $Qbigl(frac{d}{dy}bigr)$ at $x$:
      $$
      Pleft(frac{d}{dx}right)
      =sum_{n=0}^inftyfrac{P^{(n)}(y)}{n!}left(frac{d}{dx} - yright)^n,
      quad
      Qleft(frac{d}{dy}right)
      =sum_{n=0}^inftyfrac{Q^{(n)}(x)}{n!}left(frac{d}{dy} - xright)^n.
      $$



      Is there any easy way to "see" that
      $Pbigl(frac{d}{dx}bigr)bigl(e^{xy}Q(x)) = Qbigl(frac{d}{dy}bigr)bigl(e^{xy}P(y)bigr)$
      without "getting hands dirty"?



      Is this identity a part of some general theory?
      It makes me think of Fourier analysis, but I do not know much about it.







      ordinary-differential-equations polynomials exponential-function differential-operators






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




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      edited Dec 3 '18 at 12:56







      Alexey

















      asked Dec 3 '18 at 7:53









      AlexeyAlexey

      756623




      756623






















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