{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,4]],"date-time":"2022-04-04T23:45:45Z","timestamp":1649115945187},"reference-count":10,"publisher":"Hindawi Limited","license":[{"start":{"date-parts":[[2007,1,1]],"date-time":"2007-01-01T00:00:00Z","timestamp":1167609600000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["International Journal of Mathematics and Mathematical Sciences"],"published-print":{"date-parts":[[2007]]},"abstract":"<jats:p>For a noncommuting product of functions, similar to convolutions, an \u201cidentity-type function\u201d leaving a specific function invariant is defined. It is evaluated for any choice of function on which it acts by solving a functional equation. A closed-form representation for the identity-type function of<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mrow><mml:msup><mml:mrow><mml:mrow><mml:mo>(<\/mml:mo><mml:mrow><mml:mn>1<\/mml:mn><mml:mo>+<\/mml:mo><mml:mi>t<\/mml:mi><\/mml:mrow><mml:mo>)<\/mml:mo><\/mml:mrow><\/mml:mrow><mml:mrow><mml:mo>\u2212<\/mml:mo><mml:mi>b<\/mml:mi><\/mml:mrow><\/mml:msup><mml:mrow><mml:mo>(<\/mml:mo><mml:mrow><mml:mi>b<\/mml:mi><mml:mo>&gt;<\/mml:mo><mml:mn>0<\/mml:mn><\/mml:mrow><mml:mo>)<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:math>is obtained, which is a solution of a second-order linear differential equation with given boundary conditions. It yields orthogonal polynomials whose graphs are also given. The relevance for solution of boundary value problems by a series and convergence of the series are briefly discussed.<\/jats:p>","DOI":"10.1155\/2007\/94204","type":"journal-article","created":{"date-parts":[[2007,5,10]],"date-time":"2007-05-10T08:22:07Z","timestamp":1178785327000},"page":"1-10","source":"Crossref","is-referenced-by-count":0,"title":["Identity-Type Functions and Polynomials"],"prefix":"10.1155","volume":"2007","author":[{"given":"M. Aslam","family":"Chaudhry","sequence":"first","affiliation":[]},{"given":"Asghar","family":"Qadir","sequence":"additional","affiliation":[]}],"member":"98","reference":[{"key":"1","volume-title":"Integral Transforms and Their Applications","year":"1995"},{"key":"2","volume-title":"Tables of Integral Transforms. Vol. I","year":"1954"},{"key":"4","volume-title":"The Theory of the Riemann Zeta-Function","year":"1951"},{"key":"5","volume-title":"Table of Integrals, Series, and Products","year":"1994"},{"key":"6","volume-title":"Generalized Hypergeometric Functions","year":"1966"},{"key":"7","volume-title":"Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables","year":"1970"},{"key":"8","series-title":"Encyclopedia of Mathematics and Its Applications","volume-title":"Special Functions","volume":"71","year":"1999"},{"key":"9","volume-title":"On a Class of Incomplete Gamma Functions with Applications","year":"2002"},{"key":"11","volume-title":"An Atlas of Functions","year":"1987"},{"key":"12","series-title":"International Series in Pure and Applied Mathematics","volume-title":"Introduction to Approximation Theory","year":"1966"}],"container-title":["International Journal of Mathematics and Mathematical Sciences"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2007\/094204.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2007\/094204.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2016,8,11]],"date-time":"2016-08-11T17:57:08Z","timestamp":1470938228000},"score":1,"resource":{"primary":{"URL":"http:\/\/www.hindawi.com\/journals\/ijmms\/2007\/094204\/abs\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007]]},"references-count":10,"alternative-id":["094204","94204"],"URL":"https:\/\/doi.org\/10.1155\/2007\/94204","relation":{},"ISSN":["0161-1712","1687-0425"],"issn-type":[{"value":"0161-1712","type":"print"},{"value":"1687-0425","type":"electronic"}],"subject":[],"published":{"date-parts":[[2007]]}}}