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Paper IPM / M / 11333  


Abstract:  
In this paper, by using the extended SturmLiouville theorem for symmetric functions, we introduce the following differential equation x^{2}(1−x^{2m})Φ"_{n}(x)−2x((a+mb+1) x^{2m} −a + m − 1)Φ_{n}^{′}(x)+ (α_{n} x^{2m} + β+ [(1−(−1)^{n})/2] γ)Φ_{n}(x) = 0, in which β = −2s(2s + 2a − 2m + 1); γ = 2s(2s + 2a − 2m + 1)−2(2r+1)(r+a−m+1) and α_{n} = (mn+2s+(r−s+(m−1)/2)(1−(−1)^{n})) (mn+2s+2a+1+2mb+(r−s+(m−1)/2)(1−(−1)^{n})) and show that one of this basic solutions is a class of incomplete symmetric polynomials orthogonal with respect to the weight function x^{2a} (1−x^{2m})^{b} on [−1,1]. We also obtain the norm square value of this orthogonal class. Download TeX format 

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