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Where Q(x) terminating (1 _ "" - yfr 1) \ 2 X* H 1 \ > odd - z 2 ) = 2z a polynomial series degenerates to solution of 4-38 is 4fi . 4-65 1 therefore + 5Q^(o;) APJTCc) 4-66 represents the series which does not terminate. The must, of course, be the polynomial F(x) which we series have already studied. ^Hh? + + 2 ~"(" + l)s x + The complete u = n(n , either the even series or the of degree ~ 1) 1) Since + 2(m + b, fr 1> - n(n + l)fr + + 2) + m(m + 1) 1) lim the series for QJ* converges for the series becomes divergent.

Thus we may define an operator | as follows multiply the function by the inde- x and multiply it : variable. pendent This rule written symbolically is = |/(z) Another operator, 349 xf(x} 8, is differentiation with respect to the independent variable: = /(*) 3-20 /'(*) We may develop an algebra of operators, just as we can develop an numbers. The sum of two operators a and p is defined by the equation algebra of (a The product of = + p)/(z) two operators is a/(z) defined + p/(x) 3-21 by the equation 3-22 The resemblance ficial.

M (6) If M is one of the p'& the operator 9 is is multiplication where by the qi is con- jugate to p^ (c) If the the M , p's, g's, M any dynamical variable expressible in terms of the q's, t, the operator is found by substituting the operators for the p's, and t as defined above in the algebraic expression for is and and replacing the algebra. If there is processes of ordinary algebra by those of operator any ambiguity in the order of the factors, they must be arranged so that the resulting operator is Hermitian.

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