A NOTE ON THE HYPERGEOMETRIC AND BESSEL'S EQUATIONS ( II ) .sx By J.L. BURCHNALL ( Durham ) and T.W. CHAUNDY ( Oxford ) .sx Received 26 May 1931 .sx 1 .sx Introduction .sx IN part I of this Note we called attention to certain elementary propositions in the theory of the generalized hypergeometric equation , ( 1 ) which enable us to transform equations of this type , by substitutions of the form , ( 2 ) into equations of similar or simpler character in z. .sx Our arguments were based on the main theorem I ( 2 ) , namely :sx If z is a solution of the equation , then is a solution of the equation .sx In part I we applied this theorem to a special class of equations ( 1 ) , namely , those of the generalized Bessel's type , ( 3 ) employing it mainly to throw light on the classification of such equations by the types of functions entering into their general solution .sx In this part II we undertake a similar inquiry for the more general equations ( 1 ) , in which , however , some of the simplicity of the earlier inquiry will be lacking for the following reason .sx The substitution ( 2 ) derives the complete solution of the y-equation from the z-equation is less than that of the y-equation , or else a solution of the z-equation is annihilated by the operator .sx With equations of the generalised Bessel's type ( 3 ) neither of these possibilities occur .sx On the one hand the order of the y-equation is the difference of the orders of the o-operators on the two sides of the equation , which is invariant for substitutions of the type ( 2) .sx The z-equation cannot therefore be of lower order than the y-equation , and in cases of interest is of the same order , when consequently it is also of the generalized Bessel's form ( 3) .sx Therefore , on the other hand , no solution of the z-equation can be annihilated by the operator , for this operator annihilates only finite sums of the form , whereas the generalized Bessel's equations have no finite solutions of this form .sx Thus , as we have found in part I , the complete solution of the z-equation leads at once to the complete solution of the y-equation .sx With equations of the more general type ( 1 ) this need not always be the case , and we must then fill in the 'missing' solution ( or solutions ) from other sources .sx 2 .sx Amplification of Theorem I ( 2 ) .sx We begin by extending the theorem I ( 2 ) into a form which takes account of these missing solutions .sx We evidently lack a solution y , if and only if some z is a finite sum of powers .sx For this the necessary and sufficient condition is that have a factor , where s has one of the above values .sx If there are several such factors , suppose is that with greatest s , and consider the equation .sx ( 4 ) .sx The substitution now gives for z the equation or sufficiently , if , .sx In this case , then , as compensation for missing a solution , we have been able to lower the order of the equation .sx To replace the missing solution we observe that the full equation for z1 is actually , or sufficiently , , which has the first integral , ( 5 ) where C is some constant .sx The substitution connection y , z is now , ( 6 ) and z1 is annihilated by the operator on the right , only if it be a sum of powers .sx If such as z1 be a solution of ( 5 ) , then , whether C be zero or not , the last of the above powers must be annihilated by .sx Thus &formula must have had a factor with , which is contrary to our hypothesis about q. Consequently the complete solution of ( 5 ) with gives , through ( 6 ) , the complete solution of ( 4) .sx Replacing by we may enunciate this result as follows :sx ( 1 ) If q , r , s be integers such q r and have no factor where q s r , then the complete solution of the equation is given by , where z is the complete solution of the equation and C is a constant other than zero .sx If , themselves have each a factor of the type , where q , r are subject to the conditions stated in ( 1 ) for q , r , then we can once again apply the procedure of ( 1 ) to diminish the order of the equation , adding a further term independent of z on the right .sx The term already on the right is replaced by some X defined by an equation .sx In general it is enough to write .sx But , if the operator on the left contain a factor , we have to take X of the form .sx We can clearly proceed in this way so long as , possess corresponding factors of the requisite forms .sx We can replace ( 1 ) by the more general theorem :sx ( 2 ) If q , r , s be integers such that have no factor where , and if p be any positive integer , then the complete solution of the equation is given by , where z is the complete solution of the equation and C is a constant other than zero .sx We suppress the proof , since no new principle arises .sx There is a theorem , complementary to ( 1) :sx ( 3 ) If r is a positive integer and z any solution of the equation , then is a solution of the equation .sx In this case the order of the z-equation exceeds by unity that of the y-equation , and one solution z must be annihilated by the operator .sx We see from the form of the z-equation that it has a finite solution led by and terminated by .sx This is evidently the solution annihilated .sx In the same way that ( 2 ) is the generalization of ( 1 ) , so we have the generalization of ( 3) :sx ( 4 ) If r , p are positive integers and z is any solution of the equation , then is a solution of the equation .sx 3 .sx Equations soluble by elementary functions .sx In analogy with I ( II ) we now consider the equation , where the order of the -operators on each side is equal to the index of xm and the differences , are all integers incongruent to modulus m. We may write this equation more conveniently as , ( 6 ) where Nr , are integers of either sign or zero , and a , b are any real numbers .sx By I ( 4 ) , this is satisfied by some , where z is a solution of , ( 8 ) and N , N are the algebraically least integers of the sets ( Nr ) , ( Nr) .sx By I ( 6 ) this equation is satisfied by the solution of , and therefore ( 8 ) has the m independent solutions .sx ( 9 ) .sx Unless be a negative integer or zero , the expressions ( 9 ) for z are not expressible as finite sums of powers of x and therefore are not annihilated by the operator .sx We can thus assert that .sx ( 5 ) In general , the equation ( 7 ) is soluble by algebraic functions .sx In the exception case , when is a negative integer or zero , the substitution in ( 7 ) and a cyclic rearrangement of factors on the right make it enough to consider , in place of ( 7 ) , the equation .sx ( 10 ) .sx The exceptional case is now that in which .sx If we write the right-hand side of ( 10 ) in the form and apply ( 1 ) , we see that solutions are 'missing' only when .sx Let us divide the integers into two groups ( s ) , ( t ) , according as .sx By ( 1 ) and the subsequent discussion it appears that the complete solution of ( 10 ) is given by , where z is the complete solution of .sx ( 11 ) .sx No logarithms appear , since the factors that have been removed from the right are mutually incongruent to modulus m. .sx Operation on ( 11 ) with the operator gives the equation , that is to say , .sx ( 12 ) .sx No terms have been lost on the right , since .sx The complete solution of ( 12 ) thus includes the complete solution of ( 11) .sx In ( 12 ) the 'complementary function , is a sum of powers of x ; a 'particular integral' evidently introduces , besides powers of x , the logarithms , where m = 1 .sx Terms involving log x could also conceivably appear in the particular integral .sx In this case z must include a term which gives rise to a term on the right of ( 12) .sx But there are no such terms , since t , m-s are never congruent to modulus m. Thus no logarithms appear except those of , and the complete solution of ( 12 ) involves at most these and a finite set of power of x. The same is true of ( 7 ) in the exceptional case , and we may thus sum up and say .sx ( 6 ) The differential equation , where Nr , Nr are integers of either sign or zero , and a , b are any real numbers , is , in general , soluble in algebraic functions , and in any case is soluble in algebraic functions together with the logarithms , where m = 1 .sx 4 .sx Legendre's equation .sx In illustration we may apply the preceding theory to a discussion of Legendre's equation ( 13 ) in the case in which n is a positive integer .sx If in ( 2 ) we write , , , , , , we obtain the solutions , , recognizable as constant multiples of , .sx Since n , -(n+1 ) are of opposite parity , Theorem ( 6 ) is applicable to Legendre's equation , which , as is otherwise known , is soluble in terms of algebraic functions together with the logarithms .sx Various formulae expressing more or less explicitly in algebraic and logarithmic parts may be obtained by transforming Legendre's equation into the symbolic form , ( 14 ) and suitably manipulating the fractional operator .sx If we apply ( 4 ) to ( 13 ) , we obtain the known formula , determining the numerical factor from a comparison of leading coefficients .sx We may obtain certain less usual formlae for Pn , Qn , if we separate the cases of n even , n odd .sx Write first n=2p and in ( 1 ) set , , , , .sx We then have the complete solution of Legendre's equation , , where z is the complete solution of the equation , ( 15 ) and so .sx ( 16 ) .sx From the coefficient of C we obtain , on comparison of leading coefficients , the formulae , which may also be written as , .sx We may express these in the form of contour-integrals as , , where in both integrals is a contour in the z-plane which encloses z=x but z=0 .sx These contour-integrals are evidently special cases of those which represent the hypergeometric function .sx Again from the coefficient of C in ( 16 ) we similarly obtain , .sx If n still equals 2p and we apply ( 3 ) to ( 13 ) , we see that , if y is any solution of Legendre's equation of order 2p , then ( 17 ) is a solution of .sx ( 18 ) .sx Since ( 18 ) is satisfied by and P2p is annihilated by the operator on the right of ( 17 ) , we have , on comparison of leading coefficients , , which may be written in the equivalent forms , .sx Corresponding results for are , .sx , , , .sx In similar fashion we can obtain formulae for the associated Legendre functions by considering the differential equation satisfied by .sx SOME SELF-RECIPROCAL FUNCTIONS .sx By G. N. WATSON ( Birmingham ) .sx Received 6 July 1931 .sx 1 .sx THIS NOTE contains a few minor developments of a recent paper entitled 'Self-reciprocal functions' by Hardy and Titchmarsh .sx The problems which I discuss have arisen out of a question which was put to me by Mr. E. G. Phillips ; he had encountered for formula , ( 1 ) where p is the greatest integer such that and the dash indicates that the first term of the sum is to be halved ; and his question was whether an analogous formula existed for the series obtained from ( 1 ) by changing into .sx In this paper I first consider the more general series ( 2 ) and I transform it into an infinite series which is a natural modification of the sum on the right in ( 1 ) ; it is then a trivial matter to obtain the transformation in the special case by a limiting process .sx 6 .sx