A193685 → Pₙ + Pₙ₊₁

nonnegative coefficients

Proved for all n: all roots real what this means

Recurrence

\(P_{n}(x) = \left(5 + x\right)\,P_{n-1}(x) + \left(x\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 6 + x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	P_n(1)
0	6	1								7
1	30	12	1							43
2	150	102	19	1						272
3	750	762	235	27	1					1775
4	3750	5322	2407	451	36	1				11967
5	18750	35682	22171	6015	775	46	1			83440
6	93750	232842	190879	70291	12990	1235	57	1		602045
7	468750	1490802	1568995	753207	187201	25340	1862	69	1	4496227

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-b1b9b86e187f6cb4 · revision 1 · JSON
Definition source: derived:obreschkoff_combination
Fingerprint: f6-dfb4cc06d90ce2b1

Observed on the stored rows ?

Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (affine: slope 1, intercept 1)
Palindromic rows: no
γ-positive (gamma_positive): no
Nonnegative coefficients (nonn): yes
Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

OEIS matches ?

No OEIS matches.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

The atlas's own rows: n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:        6,        1;
  1:       30,       12,        1;
  2:      150,      102,       19,        1;
  3:      750,      762,      235,       27,        1;
  4:     3750,     5322,     2407,      451,       36,       1;
  5:    18750,    35682,    22171,     6015,      775,      46,      1;
  6:    93750,   232842,   190879,    70291,    12990,    1235,     57,     1;
  7:   468750,  1490802,  1568995,   753207,   187201,   25340,   1862,    69,    1;
  8:  2343750,  9413562, 12473767,  7594651,  2438016,  440601,  45822,  2690,   82,  1;
  9: 11718750, 58825122, 96729931, 73230975, 29536795, 6844026, 944643, 78102, 3756, 96, 1;

%F Formula

The row polynomials satisfy P(n,x) = (5+x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 6 + x.

%t Mathematica

P[0, x_] := 6 + x;
P[n_, x_] := P[n, x] = Expand[(5 + x)*P[n - 1, x] + x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 6 + x; for(n = 1, N, my(i = n + 1); v[i] = (5 + x)*v[i - 1] + x*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Generated with polytool c48629374597.

Proved properties

Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-76715f1adaa360d7

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

obreschkoff_combination e-76715f1adaa360d7

Hermite–Kakeya–Obreschkoff theorem [Obr63; Fis06, Thm 1.43] ?
- premise (sequence): 5-Stirling numbers of the second kind.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
  - theorem_check liu_wang e-9b675b6342ad9d47 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
  - basis_map A008277 e-2a53ff5ba197ee63
    
    Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ?
    
    premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*4^(i-j).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    theorem_check liu_wang e-2d1c3ef96dc1c123 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
    
    preserver_image reversal, shift 0 e-6e49a6f9dd41e218
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Triangle of coefficients in expansion of (1+4x)^n.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    theorem_check liu_wang e-f45969f98fa0cee8 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
    
    parent_provers periodic_product e-36f1e280ef03dec1 root: theorem check
    
    Parent-row provers of polytool polytool ?
    
    preserver_image reversal, shift 0 e-8a33d9f811421624
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*4^(i-j).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    ↵ theorem_check liu_wang e-2d1c3ef96dc1c123 (expanded above)
    
    basis_map A007318 e-9f0b70d11cb8d495
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): Cube of lower triangular normalized binomial matrix.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    theorem_check liu_wang e-2786234a932620b2 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
    
    preserver_image reversal, shift 0 e-090c65763405a203
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Triangle of coefficients in expansion of (1+3*x)^n.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    theorem_check liu_wang e-035d18f235569c81 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
    
    basis_map A013610 e-404109e50a21afd4
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    veronese_section r=2, j=0, shift 0 e-292206ad4ae86d2d
    
    Veronese sections of Pólya frequency sequences [ASW52; Edr53; AW24] polytool ?
    
    premise (sequence): basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    parent_provers periodic_product e-1991a148b9538b93 root: theorem check
    
    Parent-row provers of polytool polytool ?
    
    preserver_image mobius, shift 0 e-12303ad0e0b9f8eb
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    ↵ parent_provers periodic_product e-1991a148b9538b93 (expanded above)
    
    parent_provers periodic_product e-b7eca039219a7f53 root: theorem check
    
    Parent-row provers of polytool polytool ?
    
    preserver_image mobius, shift 0 e-7e44b66e04c497d6
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    ↵ veronese_section r=2, j=0, shift 0 e-292206ad4ae86d2d (expanded above)
    
    ↵ parent_provers periodic_product e-b7eca039219a7f53 (expanded above)
    
    preserver_image reversal, shift 0 e-b03bb7746bf562de
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Cube of lower triangular normalized binomial matrix.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    ↵ theorem_check liu_wang e-2786234a932620b2 (expanded above)
    
    parent_provers periodic_product e-58678fea8a24dd74 root: theorem check
    
    Parent-row provers of polytool polytool ?
    
    basis_map A007318 e-7b97c40e19745577
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    theorem_check liu_wang e-0d085d05819c04f2 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
    
    preserver_image reversal, shift 0 e-045f9d56b54ceaf2
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Triangle of coefficients in expansion of (1+2*x)^n.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    theorem_check liu_wang e-a6866143cc223753 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
    
    basis_map A013609 e-41b2234a78052e4f
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    ↵ veronese_section r=2, j=0, shift 0 e-292206ad4ae86d2d (expanded above)
    
    ↵ parent_provers periodic_product e-b7eca039219a7f53 (expanded above)
    
    ↵ preserver_image mobius, shift 0 e-7e44b66e04c497d6 (expanded above)
    
    gamma_polynomial scale 1, shift 0 e-4c4f74d9ba733cff
    
    Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?
    
    premise (sequence): Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    parent_provers periodic_product e-eab26daf7024ea74 root: theorem check
    
    Parent-row provers of polytool polytool ?
    
    preserver_image reversal, shift 0 e-6d15baf5e54583ee
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    ↵ theorem_check liu_wang e-0d085d05819c04f2 (expanded above)
    
    parent_provers periodic_product e-1697ba43e31b7729 root: theorem check
    
    Parent-row provers of polytool polytool ?
    
    basis_map A007318 e-4608a916413bc814
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): binomial
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    theorem_check liu_wang e-317eb4ccca60c6ac root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
    
    preserver_image reversal, shift 0 e-033cf842aad3bdf7
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): binomial
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    ↵ theorem_check liu_wang e-317eb4ccca60c6ac (expanded above)
    
    basis_map A007318 e-095baa2a9dfe188f
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    ↵ veronese_section r=2, j=0, shift 0 e-292206ad4ae86d2d (expanded above)
    
    ↵ parent_provers periodic_product e-b7eca039219a7f53 (expanded above)
    
    ↵ preserver_image mobius, shift 0 e-7e44b66e04c497d6 (expanded above)
    
    parent_provers periodic_product e-13983862a411779c root: theorem check
    
    Parent-row provers of polytool polytool ?
    
    preserver_image reversal, shift 0 e-989c95e5c893fce9
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Riordan array (1-x,x/(1-x)).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    parent_provers periodic_product e-b62074ba45057e82 root: theorem check
    
    Parent-row provers of polytool polytool ?
    
    gamma_polynomial scale 1, shift 0 e-c76adc2a99cd3dea
    
    Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?
    
    premise (sequence): Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    preserver_image reversal, shift 0 e-0e058c428dfdfeb6
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    parent_provers periodic_product e-d82c9b48e4b31a6b root: theorem check
    
    Parent-row provers of polytool polytool ?
    
    ↵ parent_provers periodic_product e-d82c9b48e4b31a6b (expanded above)
    
    preserver_image affine, shift 0 e-3a52de7742ed9314
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    ↵ veronese_section r=2, j=0, shift 0 e-292206ad4ae86d2d (expanded above)
    
    ↵ parent_provers periodic_product e-b7eca039219a7f53 (expanded above)
    
    ↵ preserver_image mobius, shift 0 e-7e44b66e04c497d6 (expanded above)
    
    ↵ basis_map A007318 e-095baa2a9dfe188f (expanded above)
    
    ↵ parent_provers periodic_product e-13983862a411779c (expanded above)
    
    ↵ preserver_image reversal, shift 0 e-989c95e5c893fce9 (expanded above)
    
    ↵ gamma_polynomial scale 1, shift 0 e-c76adc2a99cd3dea (expanded above)
    
    ↵ preserver_image affine, shift 0 e-3a52de7742ed9314 (expanded above)
    
    preserver_image affine, shift 0 e-17c52917511dc051
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): binomial
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    ↵ theorem_check liu_wang e-317eb4ccca60c6ac (expanded above)
    
    ↵ preserver_image reversal, shift 0 e-033cf842aad3bdf7 (expanded above)
    
    ↵ basis_map A007318 e-095baa2a9dfe188f (expanded above)
    
    ↵ parent_provers periodic_product e-13983862a411779c (expanded above)
    
    ↵ preserver_image reversal, shift 0 e-989c95e5c893fce9 (expanded above)
    
    ↵ gamma_polynomial scale 1, shift 0 e-c76adc2a99cd3dea (expanded above)
    
    ↵ preserver_image affine, shift 0 e-3a52de7742ed9314 (expanded above)
    
    parent_provers periodic_product e-d08b7f9df01d4f3c root: theorem check
    
    Parent-row provers of polytool polytool ?
    
    preserver_image mobius, shift 0 e-2c323256b27180b8
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): binomial
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    ↵ theorem_check liu_wang e-317eb4ccca60c6ac (expanded above)
    
    ↵ preserver_image reversal, shift 0 e-033cf842aad3bdf7 (expanded above)
    
    ↵ basis_map A007318 e-095baa2a9dfe188f (expanded above)
    
    ↵ parent_provers periodic_product e-13983862a411779c (expanded above)
    
    ↵ preserver_image reversal, shift 0 e-989c95e5c893fce9 (expanded above)
    
    ↵ gamma_polynomial scale 1, shift 0 e-c76adc2a99cd3dea (expanded above)
    
    ↵ preserver_image affine, shift 0 e-3a52de7742ed9314 (expanded above)
    
    ↵ parent_provers periodic_product e-1697ba43e31b7729 (expanded above)
    
    ↵ basis_map A007318 e-4608a916413bc814 (expanded above)
    
    ↵ preserver_image affine, shift 0 e-17c52917511dc051 (expanded above)
    
    preserver_image affine, shift 0 e-4b09f87a5a93f62b
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    ↵ theorem_check liu_wang e-0d085d05819c04f2 (expanded above)
    
    ↵ preserver_image reversal, shift 0 e-045f9d56b54ceaf2 (expanded above)
    
    ↵ parent_provers periodic_product e-1697ba43e31b7729 (expanded above)
    
    ↵ basis_map A007318 e-4608a916413bc814 (expanded above)
    
    ↵ preserver_image affine, shift 0 e-17c52917511dc051 (expanded above)
    
    parent_provers periodic_product e-bd3566bc3465f71b root: theorem check
    
    Parent-row provers of polytool polytool ?
    
    preserver_image mobius, shift 0 e-33c9dd76a325511a
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Triangle of coefficients in expansion of (1+2*x)^n.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    ↵ theorem_check liu_wang e-a6866143cc223753 (expanded above)
    
    ↵ basis_map A013609 e-41b2234a78052e4f (expanded above)
    
    ↵ gamma_polynomial scale 1, shift 0 e-4c4f74d9ba733cff (expanded above)
    
    ↵ preserver_image reversal, shift 0 e-6d15baf5e54583ee (expanded above)
    
    ↵ parent_provers periodic_product e-d08b7f9df01d4f3c (expanded above)
    
    ↵ preserver_image mobius, shift 0 e-2c323256b27180b8 (expanded above)
    
    ↵ parent_provers periodic_product e-58678fea8a24dd74 (expanded above)
    
    ↵ basis_map A007318 e-7b97c40e19745577 (expanded above)
    
    ↵ preserver_image affine, shift 0 e-4b09f87a5a93f62b (expanded above)
    
    parent_provers periodic_product e-bf59fe80a0559f3e root: theorem check
    
    Parent-row provers of polytool polytool ?
    
    preserver_image affine, shift 0 e-4db64ce997502606
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Cube of lower triangular normalized binomial matrix.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    ↵ theorem_check liu_wang e-2786234a932620b2 (expanded above)
    
    ↵ preserver_image reversal, shift 0 e-090c65763405a203 (expanded above)
    
    ↵ parent_provers periodic_product e-58678fea8a24dd74 (expanded above)
    
    ↵ basis_map A007318 e-7b97c40e19745577 (expanded above)
    
    ↵ preserver_image affine, shift 0 e-4b09f87a5a93f62b (expanded above)
    
    basis_map A013611 e-c19dc1e19e56bc04
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    ↵ veronese_section r=2, j=0, shift 0 e-292206ad4ae86d2d (expanded above)
    
    ↵ parent_provers periodic_product e-b7eca039219a7f53 (expanded above)
    
    ↵ preserver_image mobius, shift 0 e-7e44b66e04c497d6 (expanded above)
    
    preserver_image mobius, shift 0 e-db5094ba61234dec
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Triangle of coefficients in expansion of (1+3*x)^n.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    ↵ theorem_check liu_wang e-035d18f235569c81 (expanded above)
    
    ↵ basis_map A013610 e-404109e50a21afd4 (expanded above)
    
    ↵ preserver_image reversal, shift 0 e-b03bb7746bf562de (expanded above)
    
    ↵ parent_provers periodic_product e-bd3566bc3465f71b (expanded above)
    
    ↵ preserver_image mobius, shift 0 e-33c9dd76a325511a (expanded above)
    
    ↵ basis_map A007318 e-9f0b70d11cb8d495 (expanded above)
    
    ↵ parent_provers periodic_product e-bf59fe80a0559f3e (expanded above)
    
    ↵ preserver_image affine, shift 0 e-4db64ce997502606 (expanded above)
- premise (sequence): 5-Stirling numbers of the second kind.
  uses: Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
  
  no recorded property covers this requirement

Related sequences ?

No candidate equalities or related sequences.

References

[ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
[AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
[Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
[Fis06] Steve Fisk. Polynomials, roots, and interlacing. arXiv:math/0612833. arXiv
[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
[MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
[Obr63] Nikola Obreschkoff. Verteilung und berechnung der nullstellen reeller polynome. Hochschulbücher für mathematik, VEB Deutscher Verlag der Wissenschaften, 1963.