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13 Confluent Hypergeometric FunctionsKummer Functions

§13.9 Zeros

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§13.9(i) Zeros of M(a,b,z)

If a and b-a0,-1,-2,, then M(a,b,z) has infinitely many z-zeros in . When a,b the number of real zeros is finite. Let p(a,b) be the number of positive zeros. Then

13.9.1 p(a,b) =-a,
a<0, b0,
13.9.2 p(a,b) =0,
a0, b0,
13.9.3 p(a,b) =1,
a0, -1<b<0,
13.9.4 p(a,b)=-12b--12(b+1),
a0, b-1.
13.9.5 p(a,b)=-a--b,
-a-b, a<0, b<0,
13.9.6 p(a,b)=12(-b--a+1)-12(-b--a),
-b>-a>0.

The number of negative real zeros n(a,b) is given by

13.9.7 n(a,b)=p(b-a,b).

When a<0 and b>0 let ϕr, r=1,2,3,, be the positive zeros of M(a,b,x) arranged in increasing order of magnitude, and let jb-1,r be the rth positive zero of the Bessel function Jb-1(x)10.21(i)). Then

13.9.8 ϕr=jb-1,r22b-4a(1+2b(b-2)+jb-1,r23(2b-4a)2)+O(1a5),

as a- with r fixed.

Inequalities for ϕr are given in Gatteschi (1990), and identities involving infinite series of all of the complex zeros of M(a,b,x) are given in Ahmed and Muldoon (1980).

For fixed a,b the large z-zeros of M(a,b,z) satisfy

13.9.9 z=±(2n+a)πi+ln(-Γ(a)Γ(b-a)(±2nπi)b-2a)+O(n-1lnn),

where n is a large positive integer, and the logarithm takes its principal value (§4.2(i)).

Let Pα denote the closure of the domain that is bounded by the parabola y2=4α(x+α) and contains the origin. Then M(a,b,z) has no zeros in the regions Pb/a, if 0<ba; P1, if 1ab; Pα, where α=(2a-b+ab)/(a(a+1)), if 0<a<1 and ab<2a/(1-a). The same results apply for the nth partial sums of the Maclaurin series (13.2.2) of M(a,b,z).

More information on the location of real zeros can be found in Zarzo et al. (1995) and Segura (2008).

For fixed b and z in the large a-zeros of M(a,b,z) are given by

13.9.10 a=-π24z(n2+(b-32)n)-116z((b-32)2π2+43z2-8b(z-1)-4b2-3)+O(n-1),

where n is a large positive integer.

For fixed a and z in the function M(a,b,z) has only a finite number of b-zeros.

§13.9(ii) Zeros of U(a,b,z)

For fixed a and b in , U(a,b,z) has a finite number of z-zeros in the sector |phz|32π-δ(<32π). Let T(a,b) be the total number of zeros in the sector |phz|<π, P(a,b) be the corresponding number of positive zeros, and a, b, and a-b+1 be nonintegers. For the case b1

13.9.11 T(a,b)=-a+1,
a<0, Γ(a)Γ(a-b+1)>0,
13.9.12 T(a,b)=-a,
a<0, Γ(a)Γ(a-b+1)<0,
13.9.13 T(a,b)=0,
a>0,

and

13.9.14 P(a,b)=b-a-1,
a+1<b,
13.9.15 P(a,b)=0,
a+1b.

For the case b1 we can use T(a,b)=T(a-b+1,2-b) and P(a,b)=P(a-b+1,2-b).

In Wimp (1965) it is shown that if a,b and 2a-b>-1, then U(a,b,z) has no zeros in the sector |phz|12π.

Inequalities for the zeros of U(a,b,x) are given in Gatteschi (1990). See also Segura (2008).

For fixed b and z in the large a-zeros of U(a,b,z) are given by

13.9.16 a-n-2πzn-2zπ2+12b+14+z2(13-4π-2)+z-(b-1)2+144πzn+O(1n),

where n is a large positive integer.

For fixed a and z in , U(a,b,z) has two infinite strings of b-zeros that are asymptotic to the imaginary axis as |b|.