On a polynomial related to the Legendre function of the second kind - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T08:40:55Z http://mathoverflow.net/feeds/question/74478 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/74478/on-a-polynomial-related-to-the-legendre-function-of-the-second-kind On a polynomial related to the Legendre function of the second kind J. M. 2011-09-04T02:27:04Z 2011-09-05T10:03:37Z <p>The Legendre function of the second kind, $Q_n(z)$, along with the usual Legendre polynomial $P_n(z)$, are the two linearly independent solutions of the Legendre differential equation.</p> <p>$Q_n(z)$ can be expressed in <a href="http://dlmf.nist.gov/14.7.E2" rel="nofollow">the following form</a>:</p> <p>$$Q_n(z)=P_n(z)\mathrm{artanh}\,z-W_{n-1}(z)$$</p> <p>where $W_{n-1}(z)$ can be expressed either <a href="http://dlmf.nist.gov/14.7.E4" rel="nofollow">as</a></p> <p>$$W_{n-1}(z)=\sum_{k=1}^n \frac{P_{k-1}(z) P_{n-k}(z)}{k}$$</p> <p>or <a href="http://dlmf.nist.gov/14.7.E3" rel="nofollow">as</a></p> <p>$$W_{n-1}(z)=\sum_{k=0}^{n-1} \frac{(H_n-H_k)(n+k)!}{2^k (n-k)! (k!)^2} (z-1)^k$$</p> <p>where $H_k$ is the $k$-th harmonic number, $H_k=\sum\limits_{j=1}^k \frac1{j}$.</p> <p>My questions:</p> <ol> <li><p><em>Mathematica</em> returns a rather complicated expression for $W_{n-1}(z)$ involving the unknown solution of a certain recurrence (i.e. <code>DifferenceRoot[]</code>). Is there possibly a simpler form for this polynomial?</p></li> <li><p>Might there be a (hopefully simple) $n$-term recurrence that generates these polynomials?</p></li> </ol> <hr> <p><strong>Addendum</strong>:</p> <p>After staring long and hard at Pietro's answer, I feel now that my second question was sorta kinda dumb; I already knew that both Legendre functions satisfied the same difference equation, so it stands to reason that a linear combination of them should also be a solution to that recurrence.</p> <p>I now would like to expand my first question a bit: is it possible to express $W_n(z)$ as a <em>single</em> hypergeometric function (e.g. ${}_p F_q$ or some of the fancy multivariate ones), perhaps with one of the parameters being a negative integer? For instance, $P_n(z)$ is expressible as a Gaussian hypergeometric function ${}_2 F_1$ with one of the numerator parameters being a negative integer. Might there be something similar for the $W_n$?</p> <p>I would also like to consider an additional question: are the $W_n$ orthogonal polynomials with respect to some <em>weight function</em> $\omega(x)$ and an associated support interval $(a,b)$? That is, if</p> <p>$$\int_a^b\omega(t)W_j(t)W_k(t)\mathrm dt=0,\qquad j\neq k$$</p> <p>for some $\omega(x)$ and some interval $(a,b)$, what is this weight function and its support interval?</p> http://mathoverflow.net/questions/74478/on-a-polynomial-related-to-the-legendre-function-of-the-second-kind/74531#74531 Answer by Pietro Majer for On a polynomial related to the Legendre function of the second kind Pietro Majer 2011-09-04T19:02:40Z 2011-09-04T21:24:35Z <p>The generating function of the sequence $W_n(z)$ (shifted) is $$w(t,z):=(1-2zt+t^2)^{-\frac{1}{2}}\log\bigg(\frac{-z+t+(1-2zt+t^2)^\frac{1}{2}}{1-z}\bigg) =\sum_{n=1}^\infty\ W_{n-1} (z)\ t^n\ .$$ It verifies a simple linear first order differential equation: $$(1-2zt+t^2)w_t + (t-z)w=1$$ that translates into a three-term linear recurrence for the $W_n\$:</p> <p>$$(n+1)W_n=(2n+1)zW_{n-1}-nW_{n-2}\qquad$$</p> <p>with the initial contitions $W_0=1$ and $W_1:=\frac{3}{2}z\ .$ </p> <p><strong>edit</strong>. Note that one may start the above recurrence with $W_{-1}:=0$ and $W_0:=1$. Also note that, up to a shift, that recurrence is the same as the Legendre polynomials. This means that the polynomials $R_n:=W_{n-1}$ are the other linear independent solution to the recurrence of the Legendre polynomials, $$(n+1)y_{n+1}=(2n+1)zy_{n}-ny_{n-1}\ ,$$ that corresponds to the initial conditions $R_0=0$, $R_1=1$ (while $P_0=1$ and $P_1=z$). According to the notations of the general theory of orthogonal polynomials, these $R_n$ should be named "<em>Legendre polynomials of the second kind</em>" (not to be confused with the Legendre <em>functions</em> of the second kind). </p>