On Spieß’s conjecture on harmonic numbers
Let H n be the n -th harmonic number and let H n ( 2 ) be the n -th generalized harmonic number of order two. Spieß proved that for a nonnegative integer m and for t = 1 , 2 , and 3 , the sum R ( m , t ) = ∑ k = 0 n k m H k t can be represented as a polynomial in H n with polynomial coefficients in...
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| Autor*in: |
Jin, Hai-Tao [verfasserIn] |
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| Format: |
E-Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2013transfer abstract |
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| Umfang: |
4 |
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| Übergeordnetes Werk: |
Enthalten in: Electroless deposition of a Ag matrix on semiconducting one-dimensional nanostructures - Miguel, F.L. ELSEVIER, 2013transfer abstract, [S.l.] |
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| Übergeordnetes Werk: |
volume:161 ; year:2013 ; number:13 ; pages:2038-2041 ; extent:4 |
| Links: |
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| DOI / URN: |
10.1016/j.dam.2013.03.024 |
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| 245 | 1 | 0 | |a On Spieß’s conjecture on harmonic numbers |
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| 520 | |a Let H n be the n -th harmonic number and let H n ( 2 ) be the n -th generalized harmonic number of order two. Spieß proved that for a nonnegative integer m and for t = 1 , 2 , and 3 , the sum R ( m , t ) = ∑ k = 0 n k m H k t can be represented as a polynomial in H n with polynomial coefficients in n plus H n ( 2 ) multiplied by a polynomial in n . For t = 3 , we show that the coefficient of H n ( 2 ) in Spieß’s formula equals B m / 2 , where B m is the m -th Bernoulli number. Spieß further conjectured for t ≥ 4 such a summation takes the same form as for t ≤ 3 . We find a counterexample for t = 4 . However, we prove that the structure theorem of Spieß holds for the sum ∑ k = 0 n p ( k ) H k 4 when the polynomial p ( k ) satisfies a certain condition. We also give a structure theorem for the sum ∑ k = 0 n k m H k H k ( 2 ) . Our proofs rely on Abel’s lemma on summation by parts. | ||
| 520 | |a Let H n be the n -th harmonic number and let H n ( 2 ) be the n -th generalized harmonic number of order two. Spieß proved that for a nonnegative integer m and for t = 1 , 2 , and 3 , the sum R ( m , t ) = ∑ k = 0 n k m H k t can be represented as a polynomial in H n with polynomial coefficients in n plus H n ( 2 ) multiplied by a polynomial in n . For t = 3 , we show that the coefficient of H n ( 2 ) in Spieß’s formula equals B m / 2 , where B m is the m -th Bernoulli number. Spieß further conjectured for t ≥ 4 such a summation takes the same form as for t ≤ 3 . We find a counterexample for t = 4 . However, we prove that the structure theorem of Spieß holds for the sum ∑ k = 0 n p ( k ) H k 4 when the polynomial p ( k ) satisfies a certain condition. We also give a structure theorem for the sum ∑ k = 0 n k m H k H k ( 2 ) . Our proofs rely on Abel’s lemma on summation by parts. | ||
| 650 | 7 | |a Harmonic number |2 Elsevier | |
| 650 | 7 | |a Abel’s lemma |2 Elsevier | |
| 700 | 1 | |a Sun, Lisa H. |4 oth | |
| 773 | 0 | 8 | |i Enthalten in |n Elsevier |a Miguel, F.L. ELSEVIER |t Electroless deposition of a Ag matrix on semiconducting one-dimensional nanostructures |d 2013transfer abstract |g [S.l.] |w (DE-627)ELV011300345 |
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10.1016/j.dam.2013.03.024 doi GBVA2013005000018.pica (DE-627)ELV021728305 (ELSEVIER)S0166-218X(13)00171-6 DE-627 ger DE-627 rakwb eng 510 510 DE-600 070 VZ 660 VZ 333.7 610 VZ 43.12 bkl 43.13 bkl 44.13 bkl Jin, Hai-Tao verfasserin aut On Spieß’s conjecture on harmonic numbers 2013transfer abstract 4 nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier Let H n be the n -th harmonic number and let H n ( 2 ) be the n -th generalized harmonic number of order two. Spieß proved that for a nonnegative integer m and for t = 1 , 2 , and 3 , the sum R ( m , t ) = ∑ k = 0 n k m H k t can be represented as a polynomial in H n with polynomial coefficients in n plus H n ( 2 ) multiplied by a polynomial in n . For t = 3 , we show that the coefficient of H n ( 2 ) in Spieß’s formula equals B m / 2 , where B m is the m -th Bernoulli number. Spieß further conjectured for t ≥ 4 such a summation takes the same form as for t ≤ 3 . We find a counterexample for t = 4 . However, we prove that the structure theorem of Spieß holds for the sum ∑ k = 0 n p ( k ) H k 4 when the polynomial p ( k ) satisfies a certain condition. We also give a structure theorem for the sum ∑ k = 0 n k m H k H k ( 2 ) . Our proofs rely on Abel’s lemma on summation by parts. Let H n be the n -th harmonic number and let H n ( 2 ) be the n -th generalized harmonic number of order two. Spieß proved that for a nonnegative integer m and for t = 1 , 2 , and 3 , the sum R ( m , t ) = ∑ k = 0 n k m H k t can be represented as a polynomial in H n with polynomial coefficients in n plus H n ( 2 ) multiplied by a polynomial in n . For t = 3 , we show that the coefficient of H n ( 2 ) in Spieß’s formula equals B m / 2 , where B m is the m -th Bernoulli number. Spieß further conjectured for t ≥ 4 such a summation takes the same form as for t ≤ 3 . We find a counterexample for t = 4 . However, we prove that the structure theorem of Spieß holds for the sum ∑ k = 0 n p ( k ) H k 4 when the polynomial p ( k ) satisfies a certain condition. We also give a structure theorem for the sum ∑ k = 0 n k m H k H k ( 2 ) . Our proofs rely on Abel’s lemma on summation by parts. Harmonic number Elsevier Abel’s lemma Elsevier Sun, Lisa H. oth Enthalten in Elsevier Miguel, F.L. ELSEVIER Electroless deposition of a Ag matrix on semiconducting one-dimensional nanostructures 2013transfer abstract [S.l.] (DE-627)ELV011300345 volume:161 year:2013 number:13 pages:2038-2041 extent:4 https://doi.org/10.1016/j.dam.2013.03.024 Volltext GBV_USEFLAG_U GBV_ELV SYSFLAG_U SSG-OLC-PHA SSG-OPC-GGO GBV_ILN_20 GBV_ILN_22 GBV_ILN_30 GBV_ILN_40 GBV_ILN_70 GBV_ILN_130 43.12 Umweltchemie VZ 43.13 Umwelttoxikologie VZ 44.13 Medizinische Ökologie VZ AR 161 2013 13 2038-2041 4 045F 510 |
| spelling |
10.1016/j.dam.2013.03.024 doi GBVA2013005000018.pica (DE-627)ELV021728305 (ELSEVIER)S0166-218X(13)00171-6 DE-627 ger DE-627 rakwb eng 510 510 DE-600 070 VZ 660 VZ 333.7 610 VZ 43.12 bkl 43.13 bkl 44.13 bkl Jin, Hai-Tao verfasserin aut On Spieß’s conjecture on harmonic numbers 2013transfer abstract 4 nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier Let H n be the n -th harmonic number and let H n ( 2 ) be the n -th generalized harmonic number of order two. Spieß proved that for a nonnegative integer m and for t = 1 , 2 , and 3 , the sum R ( m , t ) = ∑ k = 0 n k m H k t can be represented as a polynomial in H n with polynomial coefficients in n plus H n ( 2 ) multiplied by a polynomial in n . For t = 3 , we show that the coefficient of H n ( 2 ) in Spieß’s formula equals B m / 2 , where B m is the m -th Bernoulli number. Spieß further conjectured for t ≥ 4 such a summation takes the same form as for t ≤ 3 . We find a counterexample for t = 4 . However, we prove that the structure theorem of Spieß holds for the sum ∑ k = 0 n p ( k ) H k 4 when the polynomial p ( k ) satisfies a certain condition. We also give a structure theorem for the sum ∑ k = 0 n k m H k H k ( 2 ) . Our proofs rely on Abel’s lemma on summation by parts. Let H n be the n -th harmonic number and let H n ( 2 ) be the n -th generalized harmonic number of order two. Spieß proved that for a nonnegative integer m and for t = 1 , 2 , and 3 , the sum R ( m , t ) = ∑ k = 0 n k m H k t can be represented as a polynomial in H n with polynomial coefficients in n plus H n ( 2 ) multiplied by a polynomial in n . For t = 3 , we show that the coefficient of H n ( 2 ) in Spieß’s formula equals B m / 2 , where B m is the m -th Bernoulli number. Spieß further conjectured for t ≥ 4 such a summation takes the same form as for t ≤ 3 . We find a counterexample for t = 4 . However, we prove that the structure theorem of Spieß holds for the sum ∑ k = 0 n p ( k ) H k 4 when the polynomial p ( k ) satisfies a certain condition. We also give a structure theorem for the sum ∑ k = 0 n k m H k H k ( 2 ) . Our proofs rely on Abel’s lemma on summation by parts. Harmonic number Elsevier Abel’s lemma Elsevier Sun, Lisa H. oth Enthalten in Elsevier Miguel, F.L. ELSEVIER Electroless deposition of a Ag matrix on semiconducting one-dimensional nanostructures 2013transfer abstract [S.l.] (DE-627)ELV011300345 volume:161 year:2013 number:13 pages:2038-2041 extent:4 https://doi.org/10.1016/j.dam.2013.03.024 Volltext GBV_USEFLAG_U GBV_ELV SYSFLAG_U SSG-OLC-PHA SSG-OPC-GGO GBV_ILN_20 GBV_ILN_22 GBV_ILN_30 GBV_ILN_40 GBV_ILN_70 GBV_ILN_130 43.12 Umweltchemie VZ 43.13 Umwelttoxikologie VZ 44.13 Medizinische Ökologie VZ AR 161 2013 13 2038-2041 4 045F 510 |
| allfields_unstemmed |
10.1016/j.dam.2013.03.024 doi GBVA2013005000018.pica (DE-627)ELV021728305 (ELSEVIER)S0166-218X(13)00171-6 DE-627 ger DE-627 rakwb eng 510 510 DE-600 070 VZ 660 VZ 333.7 610 VZ 43.12 bkl 43.13 bkl 44.13 bkl Jin, Hai-Tao verfasserin aut On Spieß’s conjecture on harmonic numbers 2013transfer abstract 4 nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier Let H n be the n -th harmonic number and let H n ( 2 ) be the n -th generalized harmonic number of order two. Spieß proved that for a nonnegative integer m and for t = 1 , 2 , and 3 , the sum R ( m , t ) = ∑ k = 0 n k m H k t can be represented as a polynomial in H n with polynomial coefficients in n plus H n ( 2 ) multiplied by a polynomial in n . For t = 3 , we show that the coefficient of H n ( 2 ) in Spieß’s formula equals B m / 2 , where B m is the m -th Bernoulli number. Spieß further conjectured for t ≥ 4 such a summation takes the same form as for t ≤ 3 . We find a counterexample for t = 4 . However, we prove that the structure theorem of Spieß holds for the sum ∑ k = 0 n p ( k ) H k 4 when the polynomial p ( k ) satisfies a certain condition. We also give a structure theorem for the sum ∑ k = 0 n k m H k H k ( 2 ) . Our proofs rely on Abel’s lemma on summation by parts. Let H n be the n -th harmonic number and let H n ( 2 ) be the n -th generalized harmonic number of order two. Spieß proved that for a nonnegative integer m and for t = 1 , 2 , and 3 , the sum R ( m , t ) = ∑ k = 0 n k m H k t can be represented as a polynomial in H n with polynomial coefficients in n plus H n ( 2 ) multiplied by a polynomial in n . For t = 3 , we show that the coefficient of H n ( 2 ) in Spieß’s formula equals B m / 2 , where B m is the m -th Bernoulli number. Spieß further conjectured for t ≥ 4 such a summation takes the same form as for t ≤ 3 . We find a counterexample for t = 4 . However, we prove that the structure theorem of Spieß holds for the sum ∑ k = 0 n p ( k ) H k 4 when the polynomial p ( k ) satisfies a certain condition. We also give a structure theorem for the sum ∑ k = 0 n k m H k H k ( 2 ) . Our proofs rely on Abel’s lemma on summation by parts. Harmonic number Elsevier Abel’s lemma Elsevier Sun, Lisa H. oth Enthalten in Elsevier Miguel, F.L. ELSEVIER Electroless deposition of a Ag matrix on semiconducting one-dimensional nanostructures 2013transfer abstract [S.l.] (DE-627)ELV011300345 volume:161 year:2013 number:13 pages:2038-2041 extent:4 https://doi.org/10.1016/j.dam.2013.03.024 Volltext GBV_USEFLAG_U GBV_ELV SYSFLAG_U SSG-OLC-PHA SSG-OPC-GGO GBV_ILN_20 GBV_ILN_22 GBV_ILN_30 GBV_ILN_40 GBV_ILN_70 GBV_ILN_130 43.12 Umweltchemie VZ 43.13 Umwelttoxikologie VZ 44.13 Medizinische Ökologie VZ AR 161 2013 13 2038-2041 4 045F 510 |
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10.1016/j.dam.2013.03.024 doi GBVA2013005000018.pica (DE-627)ELV021728305 (ELSEVIER)S0166-218X(13)00171-6 DE-627 ger DE-627 rakwb eng 510 510 DE-600 070 VZ 660 VZ 333.7 610 VZ 43.12 bkl 43.13 bkl 44.13 bkl Jin, Hai-Tao verfasserin aut On Spieß’s conjecture on harmonic numbers 2013transfer abstract 4 nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier Let H n be the n -th harmonic number and let H n ( 2 ) be the n -th generalized harmonic number of order two. Spieß proved that for a nonnegative integer m and for t = 1 , 2 , and 3 , the sum R ( m , t ) = ∑ k = 0 n k m H k t can be represented as a polynomial in H n with polynomial coefficients in n plus H n ( 2 ) multiplied by a polynomial in n . For t = 3 , we show that the coefficient of H n ( 2 ) in Spieß’s formula equals B m / 2 , where B m is the m -th Bernoulli number. Spieß further conjectured for t ≥ 4 such a summation takes the same form as for t ≤ 3 . We find a counterexample for t = 4 . However, we prove that the structure theorem of Spieß holds for the sum ∑ k = 0 n p ( k ) H k 4 when the polynomial p ( k ) satisfies a certain condition. We also give a structure theorem for the sum ∑ k = 0 n k m H k H k ( 2 ) . Our proofs rely on Abel’s lemma on summation by parts. Let H n be the n -th harmonic number and let H n ( 2 ) be the n -th generalized harmonic number of order two. Spieß proved that for a nonnegative integer m and for t = 1 , 2 , and 3 , the sum R ( m , t ) = ∑ k = 0 n k m H k t can be represented as a polynomial in H n with polynomial coefficients in n plus H n ( 2 ) multiplied by a polynomial in n . For t = 3 , we show that the coefficient of H n ( 2 ) in Spieß’s formula equals B m / 2 , where B m is the m -th Bernoulli number. Spieß further conjectured for t ≥ 4 such a summation takes the same form as for t ≤ 3 . We find a counterexample for t = 4 . However, we prove that the structure theorem of Spieß holds for the sum ∑ k = 0 n p ( k ) H k 4 when the polynomial p ( k ) satisfies a certain condition. We also give a structure theorem for the sum ∑ k = 0 n k m H k H k ( 2 ) . Our proofs rely on Abel’s lemma on summation by parts. Harmonic number Elsevier Abel’s lemma Elsevier Sun, Lisa H. oth Enthalten in Elsevier Miguel, F.L. ELSEVIER Electroless deposition of a Ag matrix on semiconducting one-dimensional nanostructures 2013transfer abstract [S.l.] (DE-627)ELV011300345 volume:161 year:2013 number:13 pages:2038-2041 extent:4 https://doi.org/10.1016/j.dam.2013.03.024 Volltext GBV_USEFLAG_U GBV_ELV SYSFLAG_U SSG-OLC-PHA SSG-OPC-GGO GBV_ILN_20 GBV_ILN_22 GBV_ILN_30 GBV_ILN_40 GBV_ILN_70 GBV_ILN_130 43.12 Umweltchemie VZ 43.13 Umwelttoxikologie VZ 44.13 Medizinische Ökologie VZ AR 161 2013 13 2038-2041 4 045F 510 |
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10.1016/j.dam.2013.03.024 doi GBVA2013005000018.pica (DE-627)ELV021728305 (ELSEVIER)S0166-218X(13)00171-6 DE-627 ger DE-627 rakwb eng 510 510 DE-600 070 VZ 660 VZ 333.7 610 VZ 43.12 bkl 43.13 bkl 44.13 bkl Jin, Hai-Tao verfasserin aut On Spieß’s conjecture on harmonic numbers 2013transfer abstract 4 nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier Let H n be the n -th harmonic number and let H n ( 2 ) be the n -th generalized harmonic number of order two. Spieß proved that for a nonnegative integer m and for t = 1 , 2 , and 3 , the sum R ( m , t ) = ∑ k = 0 n k m H k t can be represented as a polynomial in H n with polynomial coefficients in n plus H n ( 2 ) multiplied by a polynomial in n . For t = 3 , we show that the coefficient of H n ( 2 ) in Spieß’s formula equals B m / 2 , where B m is the m -th Bernoulli number. Spieß further conjectured for t ≥ 4 such a summation takes the same form as for t ≤ 3 . We find a counterexample for t = 4 . However, we prove that the structure theorem of Spieß holds for the sum ∑ k = 0 n p ( k ) H k 4 when the polynomial p ( k ) satisfies a certain condition. We also give a structure theorem for the sum ∑ k = 0 n k m H k H k ( 2 ) . Our proofs rely on Abel’s lemma on summation by parts. Let H n be the n -th harmonic number and let H n ( 2 ) be the n -th generalized harmonic number of order two. Spieß proved that for a nonnegative integer m and for t = 1 , 2 , and 3 , the sum R ( m , t ) = ∑ k = 0 n k m H k t can be represented as a polynomial in H n with polynomial coefficients in n plus H n ( 2 ) multiplied by a polynomial in n . For t = 3 , we show that the coefficient of H n ( 2 ) in Spieß’s formula equals B m / 2 , where B m is the m -th Bernoulli number. Spieß further conjectured for t ≥ 4 such a summation takes the same form as for t ≤ 3 . We find a counterexample for t = 4 . However, we prove that the structure theorem of Spieß holds for the sum ∑ k = 0 n p ( k ) H k 4 when the polynomial p ( k ) satisfies a certain condition. We also give a structure theorem for the sum ∑ k = 0 n k m H k H k ( 2 ) . Our proofs rely on Abel’s lemma on summation by parts. Harmonic number Elsevier Abel’s lemma Elsevier Sun, Lisa H. oth Enthalten in Elsevier Miguel, F.L. ELSEVIER Electroless deposition of a Ag matrix on semiconducting one-dimensional nanostructures 2013transfer abstract [S.l.] (DE-627)ELV011300345 volume:161 year:2013 number:13 pages:2038-2041 extent:4 https://doi.org/10.1016/j.dam.2013.03.024 Volltext GBV_USEFLAG_U GBV_ELV SYSFLAG_U SSG-OLC-PHA SSG-OPC-GGO GBV_ILN_20 GBV_ILN_22 GBV_ILN_30 GBV_ILN_40 GBV_ILN_70 GBV_ILN_130 43.12 Umweltchemie VZ 43.13 Umwelttoxikologie VZ 44.13 Medizinische Ökologie VZ AR 161 2013 13 2038-2041 4 045F 510 |
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Enthalten in Electroless deposition of a Ag matrix on semiconducting one-dimensional nanostructures [S.l.] volume:161 year:2013 number:13 pages:2038-2041 extent:4 |
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Electroless deposition of a Ag matrix on semiconducting one-dimensional nanostructures |
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on spieß’s conjecture on harmonic numbers |
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On Spieß’s conjecture on harmonic numbers |
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Let H n be the n -th harmonic number and let H n ( 2 ) be the n -th generalized harmonic number of order two. Spieß proved that for a nonnegative integer m and for t = 1 , 2 , and 3 , the sum R ( m , t ) = ∑ k = 0 n k m H k t can be represented as a polynomial in H n with polynomial coefficients in n plus H n ( 2 ) multiplied by a polynomial in n . For t = 3 , we show that the coefficient of H n ( 2 ) in Spieß’s formula equals B m / 2 , where B m is the m -th Bernoulli number. Spieß further conjectured for t ≥ 4 such a summation takes the same form as for t ≤ 3 . We find a counterexample for t = 4 . However, we prove that the structure theorem of Spieß holds for the sum ∑ k = 0 n p ( k ) H k 4 when the polynomial p ( k ) satisfies a certain condition. We also give a structure theorem for the sum ∑ k = 0 n k m H k H k ( 2 ) . Our proofs rely on Abel’s lemma on summation by parts. |
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Let H n be the n -th harmonic number and let H n ( 2 ) be the n -th generalized harmonic number of order two. Spieß proved that for a nonnegative integer m and for t = 1 , 2 , and 3 , the sum R ( m , t ) = ∑ k = 0 n k m H k t can be represented as a polynomial in H n with polynomial coefficients in n plus H n ( 2 ) multiplied by a polynomial in n . For t = 3 , we show that the coefficient of H n ( 2 ) in Spieß’s formula equals B m / 2 , where B m is the m -th Bernoulli number. Spieß further conjectured for t ≥ 4 such a summation takes the same form as for t ≤ 3 . We find a counterexample for t = 4 . However, we prove that the structure theorem of Spieß holds for the sum ∑ k = 0 n p ( k ) H k 4 when the polynomial p ( k ) satisfies a certain condition. We also give a structure theorem for the sum ∑ k = 0 n k m H k H k ( 2 ) . Our proofs rely on Abel’s lemma on summation by parts. |
| abstract_unstemmed |
Let H n be the n -th harmonic number and let H n ( 2 ) be the n -th generalized harmonic number of order two. Spieß proved that for a nonnegative integer m and for t = 1 , 2 , and 3 , the sum R ( m , t ) = ∑ k = 0 n k m H k t can be represented as a polynomial in H n with polynomial coefficients in n plus H n ( 2 ) multiplied by a polynomial in n . For t = 3 , we show that the coefficient of H n ( 2 ) in Spieß’s formula equals B m / 2 , where B m is the m -th Bernoulli number. Spieß further conjectured for t ≥ 4 such a summation takes the same form as for t ≤ 3 . We find a counterexample for t = 4 . However, we prove that the structure theorem of Spieß holds for the sum ∑ k = 0 n p ( k ) H k 4 when the polynomial p ( k ) satisfies a certain condition. We also give a structure theorem for the sum ∑ k = 0 n k m H k H k ( 2 ) . Our proofs rely on Abel’s lemma on summation by parts. |
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