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Additional resources for 6-Valent analogues of Eberhards theorem
Here we shall give only a very brief account of the basic properties of some of these functions. For more detailed information the reader inay refer to Watson's book. 1. ~I-~)J,(~). n= 0 This we substitute in (2), remark that term by term integration may be justified if I c / z I < 1 and compare the results with (1). Thus we obtain N eum an n's inte g a l representation (3) On(z) = Kz-"-' JW1[x + (x2 + Z - ~ ) ~ ] ~ + [ ( Xx z + z 2 ! 4]n 1 e-'dx 0 we = i6 % Jo \[t + ( t 2 + 1 ) % I n+ [t - ( t 2 + l)x]'"1 e - Z t d t , where n'>O and I S + a r g z l <%n.
These notations are Z v ( t )= H,,k), n v Y"(z) w = 2 2 7 cos = J,(z), R v k ) = - E,,(21, [x (v - p)isp,, ,v(z)/ir (x - K m x + x v ) ~ . 1 43 Furthermore the following functions are investigated there: - I I V ( z )= % [ J Y ( z + ) J-,,(z)], 7rOV(z)= i e JOT Lzcasd, n A v ( z ) =i’-yJo*e X”(z) = % [ J y ( z ) J - V ( ~ ) ] , cos(v+) d $ , sin(v+)d+. “-$ The last two functions are generalizations of Hansen’s integral see 7 . 1 2 ( 2 ) for the Bessel coefficients. 6. Addition theorems There are two types of expansions of Bessel functions which are known as addition theorems.
N= 0 This we substitute in (2), remark that term by term integration may be justified if I c / z I < 1 and compare the results with (1). Thus we obtain N eum an n's inte g a l representation (3) On(z) = Kz-"-' JW1[x + (x2 + Z - ~ ) ~ ] ~ + [ ( Xx z + z 2 ! 4]n 1 e-'dx 0 we = i6 % Jo \[t + ( t 2 + 1 ) % I n+ [t - ( t 2 + l)x]'"1 e - Z t d t , where n'>O and I S + a r g z l <%n. i 2 - rn - 1) ! (%z)2B-n-1/m n(n r=o In particular we have ! z -2 , 0 2 ( t )= Z-' + 4 ~ - ~ . Evidently O,(z) i s a polynomial in z - ' of degree n the following inequality IOn(z)I (8) n 2 1.