# A connection between ultraspherical and pseudo-ultraspherical polynomials

Research paper by Kathy Driver, Martin E. Muldoon

Indexed on: 14 Mar '16Published on: 26 Feb '16Published in: Journal of Mathematical Analysis and Applications

#### Abstract

The pseudo-ultraspherical polynomial of degree n   is defined by <img height="21" border="0" style="vertical-align:bottom" width="173" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022247X16001840-si1.gif">C˜n(λ)(x)=(−i)nCn(λ)(ix) where <img height="21" border="0" style="vertical-align:bottom" width="54" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022247X16001840-si2.gif">Cn(λ)(x) is the ultraspherical polynomial. We derive a simple expression linking the polynomials <img height="19" border="0" style="vertical-align:bottom" width="31" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022247X16001840-si3.gif">Cn(λ) and <img height="21" border="0" style="vertical-align:bottom" width="69" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022247X16001840-si4.gif">C˜n(12−λ−n) and show how to derive various properties of the zeros of <img height="19" border="0" style="vertical-align:bottom" width="31" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022247X16001840-si5.gif">C˜n(λ) when λ<1−nλ<1−n from properties of the zeros of <img height="19" border="0" style="vertical-align:bottom" width="31" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022247X16001840-si3.gif">Cn(λ) when <img height="20" border="0" style="vertical-align:bottom" width="54" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022247X16001840-si13.gif">λ>−12.