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|Paper IPM / M / 11333||
In this paper, by using the extended Sturm-Liouville theorem for symmetric functions, we introduce the following differential equation
x2(1−x2m)Φ"n(x)−2x((a+mb+1) x2m −a + m − 1)Φn′(x)+ (αn x2m + β+ [(1−(−1)n)/2] γ)Φn(x) = 0, in which β = −2s(2s + 2a − 2m + 1); γ = 2s(2s + 2a − 2m + 1)−2(2r+1)(r+a−m+1) and αn = (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−x2m)b on [−1,1]. We also obtain the norm square value of this orthogonal class.
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