The 15 references with contexts in paper P. Ivankov L., V. Obukhov P., В. Обухов П., П. Иванков Л. (2016) “Об арифметической природе значений продифференцированных по параметру гипергеометрических функций // On Arithmetic Properties of the Values of Hypergeometric Functions Differentiated with Respect to Parameter” / spz:neicon:technomag:y:2014:i:0:p:102-113

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Shidlovskii A.B.Transtsendentnye chisla[Transcendental numbers]. Moscow, Nauka Publ., 1987. 448 p. (in Russian).
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    –‡Ì ‰Îˇ  ̄ÂÌˡ Ú‡ÍÓÈ Á‡‰‡ ̃Ë ËÒÔÓÎ ̧ÁÓ‚‡Î‡Ò ̧ ̋ÙÙÂÍÚ˂̇ˇ ÍÓÌÒÚÛÍˆËˇ ÎËÌÂÈÌ ̊ı ÔË·ÎËʇ ̨ ̆Ëı ÙÓÏ. œËÏÂÌÂÌË ÒÓ‚ÏÂÒÚÌ ̊ı ÔË·ÎËÊÂÌËÈ, Û ̃ËÚ ̊‚‡ ̨ ̆Ëı ÒÔˆËÙËÍÛ Ó‰ÌÓÓ‰ÌÓ„Ó ÒÎÛ ̃‡ˇ, ÔÓÁ‚ÓÎËÎÓ ÔÓÎÛ ̃ËÚ ̧ ÒÓÓÚ‚ÂÚÒÚ‚Û ̨ ̆ËÈ ÂÁÛÎ ̧Ú‡Ú ‚ ·ÓΠӷ ̆ÂÈ ÒËÚÛ‡ˆËË. ¬
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    [1, Á‡Ï ̃‡Ìˡ Í „·‚ 7]
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    Ô˂‰ÂÌ ̊ ÙÓÏÛÎËÓ‚ÍË ‡ÁÎË ̃Ì ̊ı ÂÁÛÎ ̧Ú‡ÚÓ‚ Ó· ‡ËÙÏÂÚË ̃ÂÒÍÓÈ ÔËӉ Á̇ ̃ÂÌËÈ ÔÓ‰ËÙÙÂÂ̈ËÓ‚‡ÌÌ ̊ı ÔÓ Ô‡‡ÏÂÚÛ „ËÔÂ„ÂÓÏÂÚË ̃ÂÒÍËı ÙÛÌ͈ËÈ. œË ̋ÚÓÏ ‚Ó ‚ÒÂı Û͇Á‡ÌÌ ̊ı Ú‡Ï ÚÂÓÂχı Ô‡‡ÏÂÚ ̊ ‡ÒÒχÚË‚‡ÂÏ ̊ı ÙÛÌ͈ËÈ ‡ˆËÓ̇Π̧Ì ̊ (ÒÏ.

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    ŒˆÂÌ͇ Á̇ÏÂ̇ÚÂΡ ‰Ó·Ë N2−1∏ x=0 (x−σ)2 N1 ÓÒÌÓ‚ ̊‚‡ÂÚÒˇ ̇ Ò‡‚ÌÂÌËË ÒÚÂÔÂÌÂÈ, ‚ ÍÓÚÓ ̊ı ÔÓÒÚ ̊ ̃ËÒ· ‚ıÓ‰ˇÚ ‚ ̃ËÒÎËÚÂÎ ̧ Ë Á̇ÏÂ̇ÚÂÎ ̧ ̋ÚÓÈ ‰Ó·Ë. —ÓÓÚ‚ÂÚÒÚ‚Û ̨ ̆ ‡ÒÒÛʉÂÌË ˇ‚ΡÂÚÒˇ Òڇ̉‡ÚÌ ̊Ï (ÒÏ., ̇ÔËÏÂ, ‰Ó͇Á‡ÚÂÎ ̧ÒÚ‚Ó ÎÂÏÏ ̊ 2 ‚
    Exact
    [1, Ò. 186]
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    ) Ë ÔË‚Ó‰ËÚ Í ÓˆÂÌÍ ‚ˉ‡eγ3n. “‡ÍËÏ Ó·‡ÁÓÏ, ÏÓ‰ÛÎ ̧ Ó· ̆Â„Ó Ì‡ËÏÂÌ ̧ ̄Â„Ó Á̇ÏÂ̇ÚÂΡ ̃ËÒÂÎp0jνÓˆÂÌË‚‡ÂÚÒˇ Ò‚ÂıÛ ‚ÂÎË ̃ËÌÓÈeγ4n. œË ÔÓÎÛ ̃ÂÌËË ÓˆÂÌÍË ‰Îˇ Ó· ̆Â„Ó Ì‡ËÏÂÌ ̧ ̄Â„Ó Á̇ÏÂ̇ÚÂΡ ÍÓ ̋ÙÙˈËÂÌÚÓ‚(21)ÔËıÓ‰ËÚÒˇ Û ̃ËÚ ̊‚‡Ú ̧ ÔÓÒΉÒڂˡ ‰ËÙÙÂÂ̈ËÓ‚‡Ìˡ.

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Belogrivov I.I. Transcendence and algebraic independence of values of some E-functions. Vestnik MGU. Ser. 1. Matematika. Mekhanika, 1967, no. 2, pp. 55{62. (in Russian).
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    ̨ ̆Ëı ÒÔˆËÙËÍÛ Ó‰ÌÓÓ‰ÌÓ„Ó ÒÎÛ ̃‡ˇ, ÔÓÁ‚ÓÎËÎÓ ÔÓÎÛ ̃ËÚ ̧ ÒÓÓÚ‚ÂÚÒÚ‚Û ̨ ̆ËÈ ÂÁÛÎ ̧Ú‡Ú ‚ ·ÓΠӷ ̆ÂÈ ÒËÚÛ‡ˆËË. ¬ [1, Á‡Ï ̃‡Ìˡ Í „·‚ 7] Ô˂‰ÂÌ ̊ ÙÓÏÛÎËÓ‚ÍË ‡ÁÎË ̃Ì ̊ı ÂÁÛÎ ̧Ú‡ÚÓ‚ Ó· ‡ËÙÏÂÚË ̃ÂÒÍÓÈ ÔËӉ Á̇ ̃ÂÌËÈ ÔÓ‰ËÙÙÂÂ̈ËÓ‚‡ÌÌ ̊ı ÔÓ Ô‡‡ÏÂÚÛ „ËÔÂ„ÂÓÏÂÚË ̃ÂÒÍËı ÙÛÌ͈ËÈ. œË ̋ÚÓÏ ‚Ó ‚ÒÂı Û͇Á‡ÌÌ ̊ı Ú‡Ï ÚÂÓÂχı Ô‡‡ÏÂÚ ̊ ‡ÒÒχÚË‚‡ÂÏ ̊ı ÙÛÌ͈ËÈ ‡ˆËÓ̇Π̧Ì ̊ (ÒÏ. ‡·ÓÚ ̊
    Exact
    [2,3,4,5,6,7,8,9,10]
    Suffix
    ). ¬ [11], ÔÓ-‚ˉËÏÓÏÛ, ‚ÔÂ‚ ̊ ‰Ó͇Á‡Ì‡ ÚÂÓÂχ, ‚ ÍÓÚÓÓÈ ÙË„ÛËÛÂÚ ÔÓ‰ËÙÙÂÂ̈ËÓ‚‡Ì̇ˇ ÔÓ Ô‡‡ÏÂÚÛ ÙÛÌÍˆËˇ, Ë ̋ÚÓÚ Ô‡‡ÏÂÚ ÔËÌËχÂÚ Ë‡ˆËÓ̇Π̧ÌÓ Á̇ ̃ÂÌËÂ. ŒÚÎË ̃Ë ÔÓÒΉÌÂÈ ÛÔÓÏˇÌÛÚÓÈ ÚÂÓÂÏ ̊ ÓÚ Ô˂‰ÂÌÌÓÈ ÌËÊ ÚÂÓÂÏ ̊ 1 Á‡ÍÎ ̨ ̃‡ÂÚÒˇ ‚ ÚÓÏ, ̃ÚÓ ‚ ÚÂÓÂÏ ËÁ [11] ÙË„ÛËÛÂÚ „ËÔÂ„ÂÓÏÂÚË ̃ÂÒ͇ˇ ÙÛÌÍˆËˇ ÚËÔ‡ (1), Û ÍÓÚÓÓÈb(x) =x, Ë Á̇ ̃ÂÌˡ ÙÛÌ͈ËÈ ·ÂÛÚÒˇ Ì ‚ ÔÓËÁ‚ÓÎ ̧ÌÓÈ Ú

3
Shidlovskii A.B. Transcendence and algebraic independence of the values of entire functions of certain classes.Uchenye zapiski MGU. Vyp. 186. Matematika. T. 9. Moscow, MSU Publ., 1959, pp. 11{70. (in Russian).
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    ̨ ̆Ëı ÒÔˆËÙËÍÛ Ó‰ÌÓÓ‰ÌÓ„Ó ÒÎÛ ̃‡ˇ, ÔÓÁ‚ÓÎËÎÓ ÔÓÎÛ ̃ËÚ ̧ ÒÓÓÚ‚ÂÚÒÚ‚Û ̨ ̆ËÈ ÂÁÛÎ ̧Ú‡Ú ‚ ·ÓΠӷ ̆ÂÈ ÒËÚÛ‡ˆËË. ¬ [1, Á‡Ï ̃‡Ìˡ Í „·‚ 7] Ô˂‰ÂÌ ̊ ÙÓÏÛÎËÓ‚ÍË ‡ÁÎË ̃Ì ̊ı ÂÁÛÎ ̧Ú‡ÚÓ‚ Ó· ‡ËÙÏÂÚË ̃ÂÒÍÓÈ ÔËӉ Á̇ ̃ÂÌËÈ ÔÓ‰ËÙÙÂÂ̈ËÓ‚‡ÌÌ ̊ı ÔÓ Ô‡‡ÏÂÚÛ „ËÔÂ„ÂÓÏÂÚË ̃ÂÒÍËı ÙÛÌ͈ËÈ. œË ̋ÚÓÏ ‚Ó ‚ÒÂı Û͇Á‡ÌÌ ̊ı Ú‡Ï ÚÂÓÂχı Ô‡‡ÏÂÚ ̊ ‡ÒÒχÚË‚‡ÂÏ ̊ı ÙÛÌ͈ËÈ ‡ˆËÓ̇Π̧Ì ̊ (ÒÏ. ‡·ÓÚ ̊
    Exact
    [2,3,4,5,6,7,8,9,10]
    Suffix
    ). ¬ [11], ÔÓ-‚ˉËÏÓÏÛ, ‚ÔÂ‚ ̊ ‰Ó͇Á‡Ì‡ ÚÂÓÂχ, ‚ ÍÓÚÓÓÈ ÙË„ÛËÛÂÚ ÔÓ‰ËÙÙÂÂ̈ËÓ‚‡Ì̇ˇ ÔÓ Ô‡‡ÏÂÚÛ ÙÛÌÍˆËˇ, Ë ̋ÚÓÚ Ô‡‡ÏÂÚ ÔËÌËχÂÚ Ë‡ˆËÓ̇Π̧ÌÓ Á̇ ̃ÂÌËÂ. ŒÚÎË ̃Ë ÔÓÒΉÌÂÈ ÛÔÓÏˇÌÛÚÓÈ ÚÂÓÂÏ ̊ ÓÚ Ô˂‰ÂÌÌÓÈ ÌËÊ ÚÂÓÂÏ ̊ 1 Á‡ÍÎ ̨ ̃‡ÂÚÒˇ ‚ ÚÓÏ, ̃ÚÓ ‚ ÚÂÓÂÏ ËÁ [11] ÙË„ÛËÛÂÚ „ËÔÂ„ÂÓÏÂÚË ̃ÂÒ͇ˇ ÙÛÌÍˆËˇ ÚËÔ‡ (1), Û ÍÓÚÓÓÈb(x) =x, Ë Á̇ ̃ÂÌˡ ÙÛÌ͈ËÈ ·ÂÛÚÒˇ Ì ‚ ÔÓËÁ‚ÓÎ ̧ÌÓÈ Ú

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Shidlovskii A.B. Transcendentality and algebraic independence of the values of E-functions. Trudy Moskovskogo matematicheskogo obshchestva, 1959, vol. 10, pp. 283{320. (in Russian).
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    ̨ ̆Ëı ÒÔˆËÙËÍÛ Ó‰ÌÓÓ‰ÌÓ„Ó ÒÎÛ ̃‡ˇ, ÔÓÁ‚ÓÎËÎÓ ÔÓÎÛ ̃ËÚ ̧ ÒÓÓÚ‚ÂÚÒÚ‚Û ̨ ̆ËÈ ÂÁÛÎ ̧Ú‡Ú ‚ ·ÓΠӷ ̆ÂÈ ÒËÚÛ‡ˆËË. ¬ [1, Á‡Ï ̃‡Ìˡ Í „·‚ 7] Ô˂‰ÂÌ ̊ ÙÓÏÛÎËÓ‚ÍË ‡ÁÎË ̃Ì ̊ı ÂÁÛÎ ̧Ú‡ÚÓ‚ Ó· ‡ËÙÏÂÚË ̃ÂÒÍÓÈ ÔËӉ Á̇ ̃ÂÌËÈ ÔÓ‰ËÙÙÂÂ̈ËÓ‚‡ÌÌ ̊ı ÔÓ Ô‡‡ÏÂÚÛ „ËÔÂ„ÂÓÏÂÚË ̃ÂÒÍËı ÙÛÌ͈ËÈ. œË ̋ÚÓÏ ‚Ó ‚ÒÂı Û͇Á‡ÌÌ ̊ı Ú‡Ï ÚÂÓÂχı Ô‡‡ÏÂÚ ̊ ‡ÒÒχÚË‚‡ÂÏ ̊ı ÙÛÌ͈ËÈ ‡ˆËÓ̇Π̧Ì ̊ (ÒÏ. ‡·ÓÚ ̊
    Exact
    [2,3,4,5,6,7,8,9,10]
    Suffix
    ). ¬ [11], ÔÓ-‚ˉËÏÓÏÛ, ‚ÔÂ‚ ̊ ‰Ó͇Á‡Ì‡ ÚÂÓÂχ, ‚ ÍÓÚÓÓÈ ÙË„ÛËÛÂÚ ÔÓ‰ËÙÙÂÂ̈ËÓ‚‡Ì̇ˇ ÔÓ Ô‡‡ÏÂÚÛ ÙÛÌÍˆËˇ, Ë ̋ÚÓÚ Ô‡‡ÏÂÚ ÔËÌËχÂÚ Ë‡ˆËÓ̇Π̧ÌÓ Á̇ ̃ÂÌËÂ. ŒÚÎË ̃Ë ÔÓÒΉÌÂÈ ÛÔÓÏˇÌÛÚÓÈ ÚÂÓÂÏ ̊ ÓÚ Ô˂‰ÂÌÌÓÈ ÌËÊ ÚÂÓÂÏ ̊ 1 Á‡ÍÎ ̨ ̃‡ÂÚÒˇ ‚ ÚÓÏ, ̃ÚÓ ‚ ÚÂÓÂÏ ËÁ [11] ÙË„ÛËÛÂÚ „ËÔÂ„ÂÓÏÂÚË ̃ÂÒ͇ˇ ÙÛÌÍˆËˇ ÚËÔ‡ (1), Û ÍÓÚÓÓÈb(x) =x, Ë Á̇ ̃ÂÌˡ ÙÛÌ͈ËÈ ·ÂÛÚÒˇ Ì ‚ ÔÓËÁ‚ÓÎ ̧ÌÓÈ Ú

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Mahler K. Application of a theorem by A.B. Shidlovski.Proc. Roy. Soc. Ser. A, 1968, vol. 305, pp. 149{173.
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    ̨ ̆Ëı ÒÔˆËÙËÍÛ Ó‰ÌÓÓ‰ÌÓ„Ó ÒÎÛ ̃‡ˇ, ÔÓÁ‚ÓÎËÎÓ ÔÓÎÛ ̃ËÚ ̧ ÒÓÓÚ‚ÂÚÒÚ‚Û ̨ ̆ËÈ ÂÁÛÎ ̧Ú‡Ú ‚ ·ÓΠӷ ̆ÂÈ ÒËÚÛ‡ˆËË. ¬ [1, Á‡Ï ̃‡Ìˡ Í „·‚ 7] Ô˂‰ÂÌ ̊ ÙÓÏÛÎËÓ‚ÍË ‡ÁÎË ̃Ì ̊ı ÂÁÛÎ ̧Ú‡ÚÓ‚ Ó· ‡ËÙÏÂÚË ̃ÂÒÍÓÈ ÔËӉ Á̇ ̃ÂÌËÈ ÔÓ‰ËÙÙÂÂ̈ËÓ‚‡ÌÌ ̊ı ÔÓ Ô‡‡ÏÂÚÛ „ËÔÂ„ÂÓÏÂÚË ̃ÂÒÍËı ÙÛÌ͈ËÈ. œË ̋ÚÓÏ ‚Ó ‚ÒÂı Û͇Á‡ÌÌ ̊ı Ú‡Ï ÚÂÓÂχı Ô‡‡ÏÂÚ ̊ ‡ÒÒχÚË‚‡ÂÏ ̊ı ÙÛÌ͈ËÈ ‡ˆËÓ̇Π̧Ì ̊ (ÒÏ. ‡·ÓÚ ̊
    Exact
    [2,3,4,5,6,7,8,9,10]
    Suffix
    ). ¬ [11], ÔÓ-‚ˉËÏÓÏÛ, ‚ÔÂ‚ ̊ ‰Ó͇Á‡Ì‡ ÚÂÓÂχ, ‚ ÍÓÚÓÓÈ ÙË„ÛËÛÂÚ ÔÓ‰ËÙÙÂÂ̈ËÓ‚‡Ì̇ˇ ÔÓ Ô‡‡ÏÂÚÛ ÙÛÌÍˆËˇ, Ë ̋ÚÓÚ Ô‡‡ÏÂÚ ÔËÌËχÂÚ Ë‡ˆËÓ̇Π̧ÌÓ Á̇ ̃ÂÌËÂ. ŒÚÎË ̃Ë ÔÓÒΉÌÂÈ ÛÔÓÏˇÌÛÚÓÈ ÚÂÓÂÏ ̊ ÓÚ Ô˂‰ÂÌÌÓÈ ÌËÊ ÚÂÓÂÏ ̊ 1 Á‡ÍÎ ̨ ̃‡ÂÚÒˇ ‚ ÚÓÏ, ̃ÚÓ ‚ ÚÂÓÂÏ ËÁ [11] ÙË„ÛËÛÂÚ „ËÔÂ„ÂÓÏÂÚË ̃ÂÒ͇ˇ ÙÛÌÍˆËˇ ÚËÔ‡ (1), Û ÍÓÚÓÓÈb(x) =x, Ë Á̇ ̃ÂÌˡ ÙÛÌ͈ËÈ ·ÂÛÚÒˇ Ì ‚ ÔÓËÁ‚ÓÎ ̧ÌÓÈ Ú

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Belogrivov I.I. Transcendentality and algebraic independence of the values of E-functions of the same class.Sibirskii matematicheskii zhurnal, 1973, vol. 14, no. 1, pp. 16{35. (in Russian).
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    ̨ ̆Ëı ÒÔˆËÙËÍÛ Ó‰ÌÓÓ‰ÌÓ„Ó ÒÎÛ ̃‡ˇ, ÔÓÁ‚ÓÎËÎÓ ÔÓÎÛ ̃ËÚ ̧ ÒÓÓÚ‚ÂÚÒÚ‚Û ̨ ̆ËÈ ÂÁÛÎ ̧Ú‡Ú ‚ ·ÓΠӷ ̆ÂÈ ÒËÚÛ‡ˆËË. ¬ [1, Á‡Ï ̃‡Ìˡ Í „·‚ 7] Ô˂‰ÂÌ ̊ ÙÓÏÛÎËÓ‚ÍË ‡ÁÎË ̃Ì ̊ı ÂÁÛÎ ̧Ú‡ÚÓ‚ Ó· ‡ËÙÏÂÚË ̃ÂÒÍÓÈ ÔËӉ Á̇ ̃ÂÌËÈ ÔÓ‰ËÙÙÂÂ̈ËÓ‚‡ÌÌ ̊ı ÔÓ Ô‡‡ÏÂÚÛ „ËÔÂ„ÂÓÏÂÚË ̃ÂÒÍËı ÙÛÌ͈ËÈ. œË ̋ÚÓÏ ‚Ó ‚ÒÂı Û͇Á‡ÌÌ ̊ı Ú‡Ï ÚÂÓÂχı Ô‡‡ÏÂÚ ̊ ‡ÒÒχÚË‚‡ÂÏ ̊ı ÙÛÌ͈ËÈ ‡ˆËÓ̇Π̧Ì ̊ (ÒÏ. ‡·ÓÚ ̊
    Exact
    [2,3,4,5,6,7,8,9,10]
    Suffix
    ). ¬ [11], ÔÓ-‚ˉËÏÓÏÛ, ‚ÔÂ‚ ̊ ‰Ó͇Á‡Ì‡ ÚÂÓÂχ, ‚ ÍÓÚÓÓÈ ÙË„ÛËÛÂÚ ÔÓ‰ËÙÙÂÂ̈ËÓ‚‡Ì̇ˇ ÔÓ Ô‡‡ÏÂÚÛ ÙÛÌÍˆËˇ, Ë ̋ÚÓÚ Ô‡‡ÏÂÚ ÔËÌËχÂÚ Ë‡ˆËÓ̇Π̧ÌÓ Á̇ ̃ÂÌËÂ. ŒÚÎË ̃Ë ÔÓÒΉÌÂÈ ÛÔÓÏˇÌÛÚÓÈ ÚÂÓÂÏ ̊ ÓÚ Ô˂‰ÂÌÌÓÈ ÌËÊ ÚÂÓÂÏ ̊ 1 Á‡ÍÎ ̨ ̃‡ÂÚÒˇ ‚ ÚÓÏ, ̃ÚÓ ‚ ÚÂÓÂÏ ËÁ [11] ÙË„ÛËÛÂÚ „ËÔÂ„ÂÓÏÂÚË ̃ÂÒ͇ˇ ÙÛÌÍˆËˇ ÚËÔ‡ (1), Û ÍÓÚÓÓÈb(x) =x, Ë Á̇ ̃ÂÌˡ ÙÛÌ͈ËÈ ·ÂÛÚÒˇ Ì ‚ ÔÓËÁ‚ÓÎ ̧ÌÓÈ Ú

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Vaananen K. On the transcendence and algebraic independence of the values of certain Efunctions.Ann. Acad. Sci. Fennicae. Ser. A: Math., 1973, vol. 537, pp. 3{15.
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    ̨ ̆Ëı ÒÔˆËÙËÍÛ Ó‰ÌÓÓ‰ÌÓ„Ó ÒÎÛ ̃‡ˇ, ÔÓÁ‚ÓÎËÎÓ ÔÓÎÛ ̃ËÚ ̧ ÒÓÓÚ‚ÂÚÒÚ‚Û ̨ ̆ËÈ ÂÁÛÎ ̧Ú‡Ú ‚ ·ÓΠӷ ̆ÂÈ ÒËÚÛ‡ˆËË. ¬ [1, Á‡Ï ̃‡Ìˡ Í „·‚ 7] Ô˂‰ÂÌ ̊ ÙÓÏÛÎËÓ‚ÍË ‡ÁÎË ̃Ì ̊ı ÂÁÛÎ ̧Ú‡ÚÓ‚ Ó· ‡ËÙÏÂÚË ̃ÂÒÍÓÈ ÔËӉ Á̇ ̃ÂÌËÈ ÔÓ‰ËÙÙÂÂ̈ËÓ‚‡ÌÌ ̊ı ÔÓ Ô‡‡ÏÂÚÛ „ËÔÂ„ÂÓÏÂÚË ̃ÂÒÍËı ÙÛÌ͈ËÈ. œË ̋ÚÓÏ ‚Ó ‚ÒÂı Û͇Á‡ÌÌ ̊ı Ú‡Ï ÚÂÓÂχı Ô‡‡ÏÂÚ ̊ ‡ÒÒχÚË‚‡ÂÏ ̊ı ÙÛÌ͈ËÈ ‡ˆËÓ̇Π̧Ì ̊ (ÒÏ. ‡·ÓÚ ̊
    Exact
    [2,3,4,5,6,7,8,9,10]
    Suffix
    ). ¬ [11], ÔÓ-‚ˉËÏÓÏÛ, ‚ÔÂ‚ ̊ ‰Ó͇Á‡Ì‡ ÚÂÓÂχ, ‚ ÍÓÚÓÓÈ ÙË„ÛËÛÂÚ ÔÓ‰ËÙÙÂÂ̈ËÓ‚‡Ì̇ˇ ÔÓ Ô‡‡ÏÂÚÛ ÙÛÌÍˆËˇ, Ë ̋ÚÓÚ Ô‡‡ÏÂÚ ÔËÌËχÂÚ Ë‡ˆËÓ̇Π̧ÌÓ Á̇ ̃ÂÌËÂ. ŒÚÎË ̃Ë ÔÓÒΉÌÂÈ ÛÔÓÏˇÌÛÚÓÈ ÚÂÓÂÏ ̊ ÓÚ Ô˂‰ÂÌÌÓÈ ÌËÊ ÚÂÓÂÏ ̊ 1 Á‡ÍÎ ̨ ̃‡ÂÚÒˇ ‚ ÚÓÏ, ̃ÚÓ ‚ ÚÂÓÂÏ ËÁ [11] ÙË„ÛËÛÂÚ „ËÔÂ„ÂÓÏÂÚË ̃ÂÒ͇ˇ ÙÛÌÍˆËˇ ÚËÔ‡ (1), Û ÍÓÚÓÓÈb(x) =x, Ë Á̇ ̃ÂÌˡ ÙÛÌ͈ËÈ ·ÂÛÚÒˇ Ì ‚ ÔÓËÁ‚ÓÎ ̧ÌÓÈ Ú

8
Vaananen K. On the algebraic independence of some E-functions related to Kummer's functions.Ann. Acad. Sci. Fennicae. Ser. A: Math., 1975, vol. 1, pp. 183{194.
Total in-text references: 1
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    ̨ ̆Ëı ÒÔˆËÙËÍÛ Ó‰ÌÓÓ‰ÌÓ„Ó ÒÎÛ ̃‡ˇ, ÔÓÁ‚ÓÎËÎÓ ÔÓÎÛ ̃ËÚ ̧ ÒÓÓÚ‚ÂÚÒÚ‚Û ̨ ̆ËÈ ÂÁÛÎ ̧Ú‡Ú ‚ ·ÓΠӷ ̆ÂÈ ÒËÚÛ‡ˆËË. ¬ [1, Á‡Ï ̃‡Ìˡ Í „·‚ 7] Ô˂‰ÂÌ ̊ ÙÓÏÛÎËÓ‚ÍË ‡ÁÎË ̃Ì ̊ı ÂÁÛÎ ̧Ú‡ÚÓ‚ Ó· ‡ËÙÏÂÚË ̃ÂÒÍÓÈ ÔËӉ Á̇ ̃ÂÌËÈ ÔÓ‰ËÙÙÂÂ̈ËÓ‚‡ÌÌ ̊ı ÔÓ Ô‡‡ÏÂÚÛ „ËÔÂ„ÂÓÏÂÚË ̃ÂÒÍËı ÙÛÌ͈ËÈ. œË ̋ÚÓÏ ‚Ó ‚ÒÂı Û͇Á‡ÌÌ ̊ı Ú‡Ï ÚÂÓÂχı Ô‡‡ÏÂÚ ̊ ‡ÒÒχÚË‚‡ÂÏ ̊ı ÙÛÌ͈ËÈ ‡ˆËÓ̇Π̧Ì ̊ (ÒÏ. ‡·ÓÚ ̊
    Exact
    [2,3,4,5,6,7,8,9,10]
    Suffix
    ). ¬ [11], ÔÓ-‚ˉËÏÓÏÛ, ‚ÔÂ‚ ̊ ‰Ó͇Á‡Ì‡ ÚÂÓÂχ, ‚ ÍÓÚÓÓÈ ÙË„ÛËÛÂÚ ÔÓ‰ËÙÙÂÂ̈ËÓ‚‡Ì̇ˇ ÔÓ Ô‡‡ÏÂÚÛ ÙÛÌÍˆËˇ, Ë ̋ÚÓÚ Ô‡‡ÏÂÚ ÔËÌËχÂÚ Ë‡ˆËÓ̇Π̧ÌÓ Á̇ ̃ÂÌËÂ. ŒÚÎË ̃Ë ÔÓÒΉÌÂÈ ÛÔÓÏˇÌÛÚÓÈ ÚÂÓÂÏ ̊ ÓÚ Ô˂‰ÂÌÌÓÈ ÌËÊ ÚÂÓÂÏ ̊ 1 Á‡ÍÎ ̨ ̃‡ÂÚÒˇ ‚ ÚÓÏ, ̃ÚÓ ‚ ÚÂÓÂÏ ËÁ [11] ÙË„ÛËÛÂÚ „ËÔÂ„ÂÓÏÂÚË ̃ÂÒ͇ˇ ÙÛÌÍˆËˇ ÚËÔ‡ (1), Û ÍÓÚÓÓÈb(x) =x, Ë Á̇ ̃ÂÌˡ ÙÛÌ͈ËÈ ·ÂÛÚÒˇ Ì ‚ ÔÓËÁ‚ÓÎ ̧ÌÓÈ Ú

9
Vaananen K. On the algebraic independence of the values of some E-functions.Ann. Acad. Sci. Fennicae. Ser. A: Math., 1975, vol. 1, pp. 93{109.
Total in-text references: 1
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    ̨ ̆Ëı ÒÔˆËÙËÍÛ Ó‰ÌÓÓ‰ÌÓ„Ó ÒÎÛ ̃‡ˇ, ÔÓÁ‚ÓÎËÎÓ ÔÓÎÛ ̃ËÚ ̧ ÒÓÓÚ‚ÂÚÒÚ‚Û ̨ ̆ËÈ ÂÁÛÎ ̧Ú‡Ú ‚ ·ÓΠӷ ̆ÂÈ ÒËÚÛ‡ˆËË. ¬ [1, Á‡Ï ̃‡Ìˡ Í „·‚ 7] Ô˂‰ÂÌ ̊ ÙÓÏÛÎËÓ‚ÍË ‡ÁÎË ̃Ì ̊ı ÂÁÛÎ ̧Ú‡ÚÓ‚ Ó· ‡ËÙÏÂÚË ̃ÂÒÍÓÈ ÔËӉ Á̇ ̃ÂÌËÈ ÔÓ‰ËÙÙÂÂ̈ËÓ‚‡ÌÌ ̊ı ÔÓ Ô‡‡ÏÂÚÛ „ËÔÂ„ÂÓÏÂÚË ̃ÂÒÍËı ÙÛÌ͈ËÈ. œË ̋ÚÓÏ ‚Ó ‚ÒÂı Û͇Á‡ÌÌ ̊ı Ú‡Ï ÚÂÓÂχı Ô‡‡ÏÂÚ ̊ ‡ÒÒχÚË‚‡ÂÏ ̊ı ÙÛÌ͈ËÈ ‡ˆËÓ̇Π̧Ì ̊ (ÒÏ. ‡·ÓÚ ̊
    Exact
    [2,3,4,5,6,7,8,9,10]
    Suffix
    ). ¬ [11], ÔÓ-‚ˉËÏÓÏÛ, ‚ÔÂ‚ ̊ ‰Ó͇Á‡Ì‡ ÚÂÓÂχ, ‚ ÍÓÚÓÓÈ ÙË„ÛËÛÂÚ ÔÓ‰ËÙÙÂÂ̈ËÓ‚‡Ì̇ˇ ÔÓ Ô‡‡ÏÂÚÛ ÙÛÌÍˆËˇ, Ë ̋ÚÓÚ Ô‡‡ÏÂÚ ÔËÌËχÂÚ Ë‡ˆËÓ̇Π̧ÌÓ Á̇ ̃ÂÌËÂ. ŒÚÎË ̃Ë ÔÓÒΉÌÂÈ ÛÔÓÏˇÌÛÚÓÈ ÚÂÓÂÏ ̊ ÓÚ Ô˂‰ÂÌÌÓÈ ÌËÊ ÚÂÓÂÏ ̊ 1 Á‡ÍÎ ̨ ̃‡ÂÚÒˇ ‚ ÚÓÏ, ̃ÚÓ ‚ ÚÂÓÂÏ ËÁ [11] ÙË„ÛËÛÂÚ „ËÔÂ„ÂÓÏÂÚË ̃ÂÒ͇ˇ ÙÛÌÍˆËˇ ÚËÔ‡ (1), Û ÍÓÚÓÓÈb(x) =x, Ë Á̇ ̃ÂÌˡ ÙÛÌ͈ËÈ ·ÂÛÚÒˇ Ì ‚ ÔÓËÁ‚ÓÎ ̧ÌÓÈ Ú

10
Mahler K.Lectures on Transcendental Numbers. Springer Berlin Heidelberg, 1976. 254 p. (Ser.Lecture Notes in Mathematics; vol. 546). DOI:10.1007/BFb0081107
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    ̨ ̆Ëı ÒÔˆËÙËÍÛ Ó‰ÌÓÓ‰ÌÓ„Ó ÒÎÛ ̃‡ˇ, ÔÓÁ‚ÓÎËÎÓ ÔÓÎÛ ̃ËÚ ̧ ÒÓÓÚ‚ÂÚÒÚ‚Û ̨ ̆ËÈ ÂÁÛÎ ̧Ú‡Ú ‚ ·ÓΠӷ ̆ÂÈ ÒËÚÛ‡ˆËË. ¬ [1, Á‡Ï ̃‡Ìˡ Í „·‚ 7] Ô˂‰ÂÌ ̊ ÙÓÏÛÎËÓ‚ÍË ‡ÁÎË ̃Ì ̊ı ÂÁÛÎ ̧Ú‡ÚÓ‚ Ó· ‡ËÙÏÂÚË ̃ÂÒÍÓÈ ÔËӉ Á̇ ̃ÂÌËÈ ÔÓ‰ËÙÙÂÂ̈ËÓ‚‡ÌÌ ̊ı ÔÓ Ô‡‡ÏÂÚÛ „ËÔÂ„ÂÓÏÂÚË ̃ÂÒÍËı ÙÛÌ͈ËÈ. œË ̋ÚÓÏ ‚Ó ‚ÒÂı Û͇Á‡ÌÌ ̊ı Ú‡Ï ÚÂÓÂχı Ô‡‡ÏÂÚ ̊ ‡ÒÒχÚË‚‡ÂÏ ̊ı ÙÛÌ͈ËÈ ‡ˆËÓ̇Π̧Ì ̊ (ÒÏ. ‡·ÓÚ ̊
    Exact
    [2,3,4,5,6,7,8,9,10]
    Suffix
    ). ¬ [11], ÔÓ-‚ˉËÏÓÏÛ, ‚ÔÂ‚ ̊ ‰Ó͇Á‡Ì‡ ÚÂÓÂχ, ‚ ÍÓÚÓÓÈ ÙË„ÛËÛÂÚ ÔÓ‰ËÙÙÂÂ̈ËÓ‚‡Ì̇ˇ ÔÓ Ô‡‡ÏÂÚÛ ÙÛÌÍˆËˇ, Ë ̋ÚÓÚ Ô‡‡ÏÂÚ ÔËÌËχÂÚ Ë‡ˆËÓ̇Π̧ÌÓ Á̇ ̃ÂÌËÂ. ŒÚÎË ̃Ë ÔÓÒΉÌÂÈ ÛÔÓÏˇÌÛÚÓÈ ÚÂÓÂÏ ̊ ÓÚ Ô˂‰ÂÌÌÓÈ ÌËÊ ÚÂÓÂÏ ̊ 1 Á‡ÍÎ ̨ ̃‡ÂÚÒˇ ‚ ÚÓÏ, ̃ÚÓ ‚ ÚÂÓÂÏ ËÁ [11] ÙË„ÛËÛÂÚ „ËÔÂ„ÂÓÏÂÚË ̃ÂÒ͇ˇ ÙÛÌÍˆËˇ ÚËÔ‡ (1), Û ÍÓÚÓÓÈb(x) =x, Ë Á̇ ̃ÂÌˡ ÙÛÌ͈ËÈ ·ÂÛÚÒˇ Ì ‚ ÔÓËÁ‚ÓÎ ̧ÌÓÈ Ú

11
Ivankov P.L. Values of hypergeometric functions that are differentiated with respect to parameter.Chebyshevskii sbornik, 2012, vol. 13, no. 2, pp. 64{70. (in Russian).
Total in-text references: 3
  1. In-text reference with the coordinate start=2764
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    „Ó ÒÎÛ ̃‡ˇ, ÔÓÁ‚ÓÎËÎÓ ÔÓÎÛ ̃ËÚ ̧ ÒÓÓÚ‚ÂÚÒÚ‚Û ̨ ̆ËÈ ÂÁÛÎ ̧Ú‡Ú ‚ ·ÓΠӷ ̆ÂÈ ÒËÚÛ‡ˆËË. ¬ [1, Á‡Ï ̃‡Ìˡ Í „·‚ 7] Ô˂‰ÂÌ ̊ ÙÓÏÛÎËÓ‚ÍË ‡ÁÎË ̃Ì ̊ı ÂÁÛÎ ̧Ú‡ÚÓ‚ Ó· ‡ËÙÏÂÚË ̃ÂÒÍÓÈ ÔËӉ Á̇ ̃ÂÌËÈ ÔÓ‰ËÙÙÂÂ̈ËÓ‚‡ÌÌ ̊ı ÔÓ Ô‡‡ÏÂÚÛ „ËÔÂ„ÂÓÏÂÚË ̃ÂÒÍËı ÙÛÌ͈ËÈ. œË ̋ÚÓÏ ‚Ó ‚ÒÂı Û͇Á‡ÌÌ ̊ı Ú‡Ï ÚÂÓÂχı Ô‡‡ÏÂÚ ̊ ‡ÒÒχÚË‚‡ÂÏ ̊ı ÙÛÌ͈ËÈ ‡ˆËÓ̇Π̧Ì ̊ (ÒÏ. ‡·ÓÚ ̊ [2,3,4,5,6,7,8,9,10]). ¬
    Exact
    [11]
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    , ÔÓ-‚ˉËÏÓÏÛ, ‚ÔÂ‚ ̊ ‰Ó͇Á‡Ì‡ ÚÂÓÂχ, ‚ ÍÓÚÓÓÈ ÙË„ÛËÛÂÚ ÔÓ‰ËÙÙÂÂ̈ËÓ‚‡Ì̇ˇ ÔÓ Ô‡‡ÏÂÚÛ ÙÛÌÍˆËˇ, Ë ̋ÚÓÚ Ô‡‡ÏÂÚ ÔËÌËχÂÚ Ë‡ˆËÓ̇Π̧ÌÓ Á̇ ̃ÂÌËÂ. ŒÚÎË ̃Ë ÔÓÒΉÌÂÈ ÛÔÓÏˇÌÛÚÓÈ ÚÂÓÂÏ ̊ ÓÚ Ô˂‰ÂÌÌÓÈ ÌËÊ ÚÂÓÂÏ ̊ 1 Á‡ÍÎ ̨ ̃‡ÂÚÒˇ ‚ ÚÓÏ, ̃ÚÓ ‚ ÚÂÓÂÏ ËÁ [11] ÙË„ÛËÛÂÚ „ËÔÂ„ÂÓÏÂÚË ̃ÂÒ͇ˇ ÙÛÌÍˆËˇ ÚËÔ‡ (1), Û ÍÓÚÓÓÈb(x) =x, Ë Á̇ ̃ÂÌˡ ÙÛÌ͈ËÈ ·ÂÛÚÒˇ Ì ‚ ÔÓËÁ‚ÓÎ ̧ÌÓÈ ÚÓ ̃Í ξ6=

  2. In-text reference with the coordinate start=3035
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    ‡·ÓÚ ̊ [2,3,4,5,6,7,8,9,10]). ¬ [11], ÔÓ-‚ˉËÏÓÏÛ, ‚ÔÂ‚ ̊ ‰Ó͇Á‡Ì‡ ÚÂÓÂχ, ‚ ÍÓÚÓÓÈ ÙË„ÛËÛÂÚ ÔÓ‰ËÙÙÂÂ̈ËÓ‚‡Ì̇ˇ ÔÓ Ô‡‡ÏÂÚÛ ÙÛÌÍˆËˇ, Ë ̋ÚÓÚ Ô‡‡ÏÂÚ ÔËÌËχÂÚ Ë‡ˆËÓ̇Π̧ÌÓ Á̇ ̃ÂÌËÂ. ŒÚÎË ̃Ë ÔÓÒΉÌÂÈ ÛÔÓÏˇÌÛÚÓÈ ÚÂÓÂÏ ̊ ÓÚ Ô˂‰ÂÌÌÓÈ ÌËÊ ÚÂÓÂÏ ̊ 1 Á‡ÍÎ ̨ ̃‡ÂÚÒˇ ‚ ÚÓÏ, ̃ÚÓ ‚ ÚÂÓÂÏ ËÁ
    Exact
    [11]
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    ÙË„ÛËÛÂÚ „ËÔÂ„ÂÓÏÂÚË ̃ÂÒ͇ˇ ÙÛÌÍˆËˇ ÚËÔ‡ (1), Û ÍÓÚÓÓÈb(x) =x, Ë Á̇ ̃ÂÌˡ ÙÛÌ͈ËÈ ·ÂÛÚÒˇ Ì ‚ ÔÓËÁ‚ÓÎ ̧ÌÓÈ ÚÓ ̃Í ξ6= 0, ‡ ‚ ÚÓ ̃Í ‚ˉ‡1/q, „‰Âq| ‰ÓÒÚ‡ÚÓ ̃ÌÓ ·ÓÎ ̧ ̄Ó ÔÓ ÏÓ‰ÛÎ ̨ ̃ËÒÎÓ. Œ·Ó· ̆ÂÌË ÔÓÎÛ ̃ÂÌÓ Á‡ Ò ̃ÂÚ ËÒÔÓÎ ̧ÁÓ‚‡Ìˡ ÒÓ‚ÏÂÒÚÌ ̊ı ÔË·ÎËÊÂÌËÈ.

  3. In-text reference with the coordinate start=11433
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    À„ÍÓ ‚ˉÂÚ ̧, ̃ÚÓ Á‰ÂÒ ̧ Ú·ÛÂÚÒˇ ‰ÓÔÓÎÌËÚÂÎ ̧ÌÓ ÓˆÂÌËÚ ̧ Ó· ̆Ë ̇ËÏÂÌ ̧ ̄Ë Á̇ÏÂ̇ÚÂÎË ̃ËÒÂÎ 1 2 , . . . , 1 N4 ,(23) ‡ Ú‡ÍÊ ̃ËÒÂÎ 1 λ+ 1 , . . . , 1 λ+N2 .(24) ’ÓÓ ̄Ó ËÁ‚ÂÒÚÌÓ, ̃ÚÓ ‰Îˇ ̃ËÒÂÎ (23) ÒÓÓÚ‚ÂÚÒÚ‚Û ̨ ̆‡ˇ ÓˆÂÌ͇ ÂÒÚ ̧eγ5n. ƒÎˇ ̃ËÒÂÎ (24) Ú·ÛÂχˇ ÓˆÂÌ͇ ÔÓÎÛ ̃Â̇ ‚
    Exact
    [11, Ò. 68{69]
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    ; Á‰ÂÒ ̧ ÏÓ‰ÛÎ ̧ Ó· ̆Â„Ó Ì‡ËÏÂÌ ̧ ̄Â„Ó Á̇ÏÂ̇ÚÂΡ ÓˆÂÌË‚‡ÂÚÒˇ Ò‚ÂıÛ ‚ÂÎË ̃ËÌÓÈeγ6(n!) u 2(2u−1). —ÛÏÏËÛˇ ‚Ò ‚ ̊ ̄ÂÒ͇Á‡ÌÌÓÂ, ÔÓÎÛ ̃‡ÂÏ Ú·ÛÂÏÛ ̨ ÓˆÂÌÍÛ. ÀÂÏχ ‰Ó͇Á‡Ì‡. ÓÏ ӈÂÌÍË Ó· ̆Â„Ó Ì‡ËÏÂÌ ̧ ̄Â„Ó Á̇ÏÂ̇ÚÂΡ ÍÓ ̋ÙÙˈËÂÌÚÓ‚ ÏÌÓ„Ó ̃ÎÂÌÓ‚ (8) ‰Îˇ ‰Ó͇Á‡ÚÂÎ ̧ÒÚ‚‡ ÓÒÌÓ‚ÌÓÈ ÚÂÓÂÏ ̊ ‚‡ÊÌÓ ËÏÂÚ ̧ Ú‡ÍÊ ӈÂÌÍË Ò‚ÂıÛ ÏÓ‰ÛÎÂÈ Á̇ ̃ÂÌËÈ (ÔË z=ξ) ̋ÚËı ÏÌÓ„Ó ̃ÎÂÌÓ‚ Ë ÏÓ‰ÛÎÂÈ Á̇ ̃ÂÌËÈ ÙÛÌ͈ËÈ r

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Ivankov P.L. On the values of some functions satisfying homogeneous differential equations. Chebyshevskii sbornik, 2013, vol. 14, no. 2, pp. 104{112. (in Russian).
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  1. In-text reference with the coordinate start=4270
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    “ÂÓÂχ 1.œÛÒÚ ̧ξ| ÌÂÌÛ΂Ӡ̃ËÒÎÓ ËÁ ÔÓΡI;hlj,l= 0,1,j= 1, . . . , u, | ÌÂÚ˂ˇΠ̧Ì ̊È Ì‡·Ó ˆÂÎ ̊ı ̃ËÒÂÎ ËÁ ̋ÚÓ„Ó ÔÓΡ;H| χÍÒËÏÛÏ ÏÓ‰ÛÎÂÈ ̋ÚËı ̃ËÒÂÎ. “Ó„‰‡ ‰Îˇ Î ̨·Ó„Óε >0̇ȉÂÚÒˇ ÔÓÎÓÊËÚÂÎ ̧ÌÓÂH0, Ú‡ÍÓÂ, ̃ÚÓ ÔËH>H0‚ ̊ÔÓÎÌˇÂÚÒˇ ÌÂ‡‚ÂÌÒÚ‚Ó∣ ∣ ∣ ∣ ∣ ∑1 l=0 ∑u j=1 hljFlj(ξ) ∣ ∣ ∣ ∣ ∣ > H1−4u−ε. 2. ƒÓ͇Á‡ÚÂÎ ̧ÒÚ‚Ó ÓÒÌÓ‚ÌÓÈ ÚÂÓÂÏ ̊ ¬ ‡·ÓÚÂ
    Exact
    [12]
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    ‡ÒÒχÚË‚‡ ̨ÚÒˇ ÙÛÌ͈ËË ‚ˉ‡ Fklkj(z) = ∑∞ ν=0 zννj−1 ∏ν x=1 a(x) b(x) dlk dλ lk k ∏ν x=1 1 x+λ , k= 1, . . . , t, lk= 0,1, . . . , τk−1, j= 1, . . . , u, „‰Âτ1, . . . ,τt| ÌÂÍÓÚÓ ̊ ̇ÚÛ‡Î ̧Ì ̊ ̃ËÒ·. ƒÎˇ ̋ÚËı ÙÛÌ͈ËÈ ‚ ÛÔÓÏˇÌÛÚÓÈ ‡·ÓÚ Ô‰ÎÓÊÂ̇ ̋ÙÙÂÍÚ˂̇ˇ ÍÓÌÒÚÛÍˆËˇ ÒÓ‚ÏÂÒÚÌ ̊ı ÔË·ÎËÊÂÌËÈ, Ú.

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    à ̊ ËÒÔÓÎ ̧ÁÛÂÏ Á‰ÂÒ ̧ ̋ÚÛ ÍÓÌÒÚÛÍˆË ̨, Ô‰‚‡ËÚÂÎ ̧ÌÓ ‰‡‚  Í‡ÚÍÓ ÓÔËÒ‡ÌË ÔËÏÂÌËÚÂÎ ̧ÌÓ Í ‡ÒÒχÚË‚‡ÂÏÓÈ ‚ ‰‡ÌÌÓÈ ‡·ÓÚ ÒËÚÛ‡ˆËË (Ú.Â. ‚ ÒÎÛ ̃‡Âk= 1,l1=l= 1,a(x) = 1). ŒÒÌӂ̇ˇ ÚÛ‰ÌÓÒÚ ̧ Á‡ÍÎ ̨ ̃‡ÂÚÒˇ ‚ ÓˆÂÌÍ ӷ ̆Â„Ó Ì‡ËÏÂÌ ̧ ̄Â„Ó Á̇ÏÂ̇ÚÂΡ ÍÓ ̋ÙÙˈËÂÌÚÓ‚ ÏÌÓ„Ó ̃ÎÂÌÓ‚Pklkj(z). ¬ ‡·ÓÚÂ
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    Ú‡ÍÓÈ ÔÓ·ÎÂÏ ̊ Ì · ̊ÎÓ, Ú‡Í Í‡Í Ú‡Ï ‡ÒÒχÚË‚‡ÂÚÒˇ ÒÎÛ ̃‡È ‡ˆËÓ̇Π̧Ì ̊ıλ1, . . . ,λt. »Ú‡Í, ÔÛÒÚ ̧n| ̇ÚÛ‡Î ̧ÌÓ ̃ËÒÎÓ, Ë ÔÛÒÚ ̧ N1= [ 2un 2u−1 ] , N2= [ un 2u−1 ] ,(2) Q(z) = N2−1∏ x=0 (z−λ+x)2,(3) θμ= 1 2πi ∫ Γ1 Q(ζ)dζ N1∏ x=N1−μ (ζ+x) , μ= 0,1, .

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    «‚ÂÁ‰Ó ̃ÍË∗Á‰ÂÒ ̧ ÔÓÒÚ‡‚ÎÂÌ ̊ ‰Îˇ ÚÓ„Ó, ̃ÚÓ· ̊ ‡ÁÎË ̃‡Ú ̧ ÏÌÓ„Ó ̃ÎÂÌP∗lj(z)Ë ‚ÒÚ ̃‡ ̨ ̆ËÈÒˇ ÌËÊ ÏÌÓ„Ó ̃ÎÂÌ (8). ŒÚÏÂÚËÏ Ú‡ÍÊÂ, ̃ÚÓ, ıÓÚˇ Û͇Á‡ÌÌ ̊ ÏÌÓ„Ó ̃ÎÂÌ ̊, Í‡Í Ë Ëı ÍÓ ̋ÙÙˈËÂÌÚ ̊, Á‡‚ËÒˇÚ ÓÚn, ÔÓ ÒÎÓÊË‚ ̄ÂÈÒˇ Ú‡‰ËˆËË ̋Ú‡ Á‡‚ËÒËÏÓÒÚ ̧ Ì Û͇Á ̊‚‡ÂÚÒˇ. ¬ ˆËÚËÓ‚‡ÌÌÓÈ ‚ ̊ ̄ ‡·ÓÚÂ
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    ‰Ó͇Á‡ÌÓ, ̃ÚÓ (ÔË ‚ ̊ÔÓÎÌÂÌËË ÛÒÎÓ‚ËÈ ÚÂÓÂÏ ̊) ÙÛÌ͈ËË P∗lj(z)Fl′j′(z)−P∗l′j′(z)Flj(z)(7) ÓÚÎË ̃Ì ̊ ÓÚ ÚÓʉÂÒÚ‚ÂÌÌÓ„Ó ÌÛΡ (ÂÒÎË(l, j)6= (l′, j′)) Ë ËÏ ̨Ú ÔËz= 0ÔÓˇ‰ÓÍ ÌÛΡ Ì ÌËÊÂN1+ 1. ƒÎˇ ̇ ̄Ëı ˆÂÎÂÈ Û‰Ó·Ì ËÒÔÓÎ ̧ÁÓ‚‡Ú ̧ ÏÌÓ„Ó ̃ÎÂÌ ̊ Plj(z) = ∑n ν=0 pljνzν,(8) „‰Â pljν=qp∗ljν,(9) q= (λ+ 1). . .(λ+N1−N2) N1! .(10) Œ ̃‚ˉÌÓ, ̃ÚÓ ÔË ̋ÚÓÏ Ò‰Â·ÌÌ ̊ ‚ ̊ ̄ ÛÚ‚ÂʉÂÌˡ Ó ÙÛÌÍˆËˇı (7) Ó

14
Ivankov P.L. On the linear independence of some functions.Chebyshevskii sbornik, 2010, vol. 11, no. 1, pp. 145{151. (in Russian).
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    ¬ÓÁÏÓÊÌÓÒÚ ̧ Ú‡ÍÓ„Ó ÔÓÒÚÓÂÌˡ ÓÒÌÓ‚‡Ì‡ ̇ ÚÓÏ, ̃ÚÓ ÙÛÌ͈ËË (1) Û‰Ó‚ÎÂÚ‚Óˇ ̨Ú ÒËÒÚÂÏ ‰ËÙÙÂÂ̈ˇΠ̧Ì ̊ı Û‡‚ÌÂÌËÈ, ‡ ÙÛÌ͈ËË (25) ËÏ ̨Ú ÔÓˇ‰ÓÍ ÌÛΡ, ·ÎËÁÍËÈ Í Ï‡ÍÒËχΠ̧ÌÓ ‚ÓÁÏÓÊÌÓÏÛ. »ÒÔÓÎ ̧ÁÛÂÚÒˇ Ú‡ÍÊÂ Ë ÎËÌÂÈ̇ˇ ÌÂÁ‡‚ËÒËÏÓÒÚ ̧ ÙÛÌ͈ËÈ (1) ̇‰ ÔÓÎÂÏ ‡ˆËÓ̇Π̧Ì ̊ı ‰Ó·ÂÈ, ÛÒÚ‡ÌÓ‚ÎÂÌ̇ˇ ‚
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    [14]
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    . œÓ‰Ó·ÌÓ Û͇Á‡ÌÌÓ ÔÓÒÚÓÂÌË ‡ÒÒχÚË‚‡ÂÚÒˇ ‚ ‡·ÓÚ [15]; ÏÂÌ ÔÓ‰Ó·ÌÓ | ‚ ‡·ÓÚ [16]. «‡ÚÂÏ ÓÒÛ ̆ÂÒڂΡÂÚÒˇ ÔÂÂıÓ‰ ÓÚ ÙÛÌ͈ËÓ̇Π̧Ì ̊ı ÔË·ÎËÊÂÌËÈ Í ̃ËÒÎÓ‚ ̊Ï. ›Ú‡ Ôӈ‰Û‡ Ú‡ÍÊ ˇ‚ΡÂÚÒˇ Òڇ̉‡ÚÌÓÈ; ÔÓ‰Ó·ÌÓ Ó̇ ÓÔË҇̇ ‚ ‡·ÓÚ [15]. œÓÎÛ ̃‡ ̨ ̆ËÈÒˇ ÔË ̋ÚÓÏ ÂÁÛÎ ̧Ú‡Ú ÓÙÓÏËÏ ‚ ‚ˉ ÎÂÏÏ ̊.

15
Chudnovsky D.V., Chudnovsky G.V. Applications of Pade approximation to Diophantine inequalities in values of G-function. In:Number Theory. Springer Berlin Heidelberg, 1985, pp. 9{51. (Ser.Lecture Notes in Mathematics; vol. 1135). DOI:10.1007/BFb0074600
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  1. In-text reference with the coordinate start=12822
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    ¬ÓÁÏÓÊÌÓÒÚ ̧ Ú‡ÍÓ„Ó ÔÓÒÚÓÂÌˡ ÓÒÌÓ‚‡Ì‡ ̇ ÚÓÏ, ̃ÚÓ ÙÛÌ͈ËË (1) Û‰Ó‚ÎÂÚ‚Óˇ ̨Ú ÒËÒÚÂÏ ‰ËÙÙÂÂ̈ˇΠ̧Ì ̊ı Û‡‚ÌÂÌËÈ, ‡ ÙÛÌ͈ËË (25) ËÏ ̨Ú ÔÓˇ‰ÓÍ ÌÛΡ, ·ÎËÁÍËÈ Í Ï‡ÍÒËχΠ̧ÌÓ ‚ÓÁÏÓÊÌÓÏÛ. »ÒÔÓÎ ̧ÁÛÂÚÒˇ Ú‡ÍÊÂ Ë ÎËÌÂÈ̇ˇ ÌÂÁ‡‚ËÒËÏÓÒÚ ̧ ÙÛÌ͈ËÈ (1) ̇‰ ÔÓÎÂÏ ‡ˆËÓ̇Π̧Ì ̊ı ‰Ó·ÂÈ, ÛÒÚ‡ÌÓ‚ÎÂÌ̇ˇ ‚ [14]. œÓ‰Ó·ÌÓ Û͇Á‡ÌÌÓ ÔÓÒÚÓÂÌË ‡ÒÒχÚË‚‡ÂÚÒˇ ‚ ‡·ÓÚÂ
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    ; ÏÂÌ ÔÓ‰Ó·ÌÓ | ‚ ‡·ÓÚ [16]. «‡ÚÂÏ ÓÒÛ ̆ÂÒڂΡÂÚÒˇ ÔÂÂıÓ‰ ÓÚ ÙÛÌ͈ËÓ̇Π̧Ì ̊ı ÔË·ÎËÊÂÌËÈ Í ̃ËÒÎÓ‚ ̊Ï. ›Ú‡ Ôӈ‰Û‡ Ú‡ÍÊ ˇ‚ΡÂÚÒˇ Òڇ̉‡ÚÌÓÈ; ÔÓ‰Ó·ÌÓ Ó̇ ÓÔË҇̇ ‚ ‡·ÓÚ [15]. œÓÎÛ ̃‡ ̨ ̆ËÈÒˇ ÔË ̋ÚÓÏ ÂÁÛÎ ̧Ú‡Ú ÓÙÓÏËÏ ‚ ‚ˉ ÎÂÏÏ ̊.

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    »ÒÔÓÎ ̧ÁÛÂÚÒˇ Ú‡ÍÊÂ Ë ÎËÌÂÈ̇ˇ ÌÂÁ‡‚ËÒËÏÓÒÚ ̧ ÙÛÌ͈ËÈ (1) ̇‰ ÔÓÎÂÏ ‡ˆËÓ̇Π̧Ì ̊ı ‰Ó·ÂÈ, ÛÒÚ‡ÌÓ‚ÎÂÌ̇ˇ ‚ [14]. œÓ‰Ó·ÌÓ Û͇Á‡ÌÌÓ ÔÓÒÚÓÂÌË ‡ÒÒχÚË‚‡ÂÚÒˇ ‚ ‡·ÓÚ [15]; ÏÂÌ ÔÓ‰Ó·ÌÓ | ‚ ‡·ÓÚ [16]. «‡ÚÂÏ ÓÒÛ ̆ÂÒڂΡÂÚÒˇ ÔÂÂıÓ‰ ÓÚ ÙÛÌ͈ËÓ̇Π̧Ì ̊ı ÔË·ÎËÊÂÌËÈ Í ̃ËÒÎÓ‚ ̊Ï. ›Ú‡ Ôӈ‰Û‡ Ú‡ÍÊ ˇ‚ΡÂÚÒˇ Òڇ̉‡ÚÌÓÈ; ÔÓ‰Ó·ÌÓ Ó̇ ÓÔË҇̇ ‚ ‡·ÓÚÂ
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    . œÓÎÛ ̃‡ ̨ ̆ËÈÒˇ ÔË ̋ÚÓÏ ÂÁÛÎ ̧Ú‡Ú ÓÙÓÏËÏ ‚ ‚ˉ ÎÂÏÏ ̊. ÀÂÏχ 7.œË Í‡Ê‰ÓÏ Ì‡ÚÛ‡Î ̧ÌÓÏn‚ ÔÓÎÂIcÛ ̆ÂÒÚ‚ÛÂÚ Ì‡·Ó ̃ËÒÂÎw (k) lj,k= 1, . . . ,2u,l= 0,1,j= 1, . . . , u, ӷ·‰‡ ̨ ̆Ëı ÒÎÂ‰Û ̨ ̆ËÏË Ò‚ÓÈÒÚ‚‡ÏË: 1) ÓÔ‰ÂÎËÚÂÎ ̧|w (k) lj|,k= 1, . . . ,2u,l= 0,1,j= 1, . . . , u, ÓÚÎË ̃ÂÌ ÓÚ ÌÛΡ; 2) ÏÓ‰ÛÎ ̧ Ó· ̆Â„Ó Ì‡ËÏÂÌ ̧ ̄Â„Ó Á̇ÏÂ̇ÚÂΡ ̃ËÒÂÎw (k) ljÌ Ô‚ ̊ ̄‡ÂÚe γ8n(n!) u 2(2u−1); 3) ∣ ∣ ∣w

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Ivankov P.L. Simultaneous Approximations with Regard to the Specific Character of the Homogeneous Case.Matematicheskie zametki, 2002, vol. 71, iss. 3, pp. 390{397. (English translation:Mathematical Notes, 2002, vol. 71, no. 3, pp. 355{361. DOI: 10.1023/A:1014898808012).
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    ¬ÓÁÏÓÊÌÓÒÚ ̧ Ú‡ÍÓ„Ó ÔÓÒÚÓÂÌˡ ÓÒÌÓ‚‡Ì‡ ̇ ÚÓÏ, ̃ÚÓ ÙÛÌ͈ËË (1) Û‰Ó‚ÎÂÚ‚Óˇ ̨Ú ÒËÒÚÂÏ ‰ËÙÙÂÂ̈ˇΠ̧Ì ̊ı Û‡‚ÌÂÌËÈ, ‡ ÙÛÌ͈ËË (25) ËÏ ̨Ú ÔÓˇ‰ÓÍ ÌÛΡ, ·ÎËÁÍËÈ Í Ï‡ÍÒËχΠ̧ÌÓ ‚ÓÁÏÓÊÌÓÏÛ. »ÒÔÓÎ ̧ÁÛÂÚÒˇ Ú‡ÍÊÂ Ë ÎËÌÂÈ̇ˇ ÌÂÁ‡‚ËÒËÏÓÒÚ ̧ ÙÛÌ͈ËÈ (1) ̇‰ ÔÓÎÂÏ ‡ˆËÓ̇Π̧Ì ̊ı ‰Ó·ÂÈ, ÛÒÚ‡ÌÓ‚ÎÂÌ̇ˇ ‚ [14]. œÓ‰Ó·ÌÓ Û͇Á‡ÌÌÓ ÔÓÒÚÓÂÌË ‡ÒÒχÚË‚‡ÂÚÒˇ ‚ ‡·ÓÚ [15]; ÏÂÌ ÔÓ‰Ó·ÌÓ | ‚ ‡·ÓÚÂ
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    . «‡ÚÂÏ ÓÒÛ ̆ÂÒڂΡÂÚÒˇ ÔÂÂıÓ‰ ÓÚ ÙÛÌ͈ËÓ̇Π̧Ì ̊ı ÔË·ÎËÊÂÌËÈ Í ̃ËÒÎÓ‚ ̊Ï. ›Ú‡ Ôӈ‰Û‡ Ú‡ÍÊ ˇ‚ΡÂÚÒˇ Òڇ̉‡ÚÌÓÈ; ÔÓ‰Ó·ÌÓ Ó̇ ÓÔË҇̇ ‚ ‡·ÓÚ [15]. œÓÎÛ ̃‡ ̨ ̆ËÈÒˇ ÔË ̋ÚÓÏ ÂÁÛÎ ̧Ú‡Ú ÓÙÓÏËÏ ‚ ‚ˉ ÎÂÏÏ ̊.

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    ; 2) ÏÓ‰ÛÎ ̧ Ó· ̆Â„Ó Ì‡ËÏÂÌ ̧ ̄Â„Ó Á̇ÏÂ̇ÚÂΡ ̃ËÒÂÎw (k) ljÌ Ô‚ ̊ ̄‡ÂÚe γ8n(n!) u 2(2u−1); 3) ∣ ∣ ∣w (k) ljFl′j′(ξ)−w (k) l′j′Flj(ξ) ∣ ∣ ∣6eγ8n(n!) −u2u−1 ; 4) ∣ ∣ ∣w (k) lj ∣ ∣ ∣6eγ8n(n!)u. ¬ ÔÓÒΉÌËı ÚÂı ÔÛÌÍÚ‡ı Ë̉ÂÍÒ ̊k,l,jÏÓ„ÛÚ ÔËÌËÏ‡Ú ̧ ‚Ò ‰ÓÔÛÒÚËÏ ̊ Á̇ ̃ÂÌˡ. »Á ÔÓÒΉÌÂÈ ÎÂÏÏ ̊ ÛÚ‚ÂʉÂÌË ÓÒÌÓ‚ÌÓÈ ÚÂÓÂÏ ̊1‚ ̊‚Ó‰ËÚÒˇ ıÓÓ ̄Ó ËÁ‚ÂÒÚÌ ̊Ï ÔËÂÏÓÏ; ÒÏ., ̇ÔËÏÂ, ‡ÒÒÛʉÂÌˡ ËÁ
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    . «‡ÍÎ ̨ ̃ÂÌË ¬ÓÁÏÓÊÌÓÒÚË ÏÂÚÓ‰‡, ËÒÔÓÎ ̧ÁÓ‚‡ÌÌÓ„Ó ‰Îˇ ‰Ó͇Á‡ÚÂÎ ̧ÒÚ‚‡ ÚÂÓÂÏ ̊1,  ̆ Ì ËÒ ̃ÂÔ‡Ì ̊. ÃÓÊÌÓ Ô‰ÔÓÎÓÊËÚ ̧, ̃ÚÓ ̋ÚÓÚ ÏÂÚÓ‰ ÔÓÁ‚ÓÎËÚ ÔÓÎÛ ̃ËÚ ̧ ‡Ì‡ÎÓ„Ë ̃Ì ̊È ÂÁÛÎ ̧Ú‡Ú Ë ‰Îˇ ‰‚‡Ê‰ ̊ ÔÓ‰ËÙÙÂÂ̈ËÓ‚‡ÌÌÓÈ ÔÓ Ô‡‡ÏÂÚÛ ÙÛÌ͈ËË ‚ˉ‡ (1). œË ̋ÚÓÏ, ÔÓ-‚ˉËÏÓÏÛ, ÔˉÂÚÒˇ Ó„‡ÌË ̃ËÚ ̧Òˇ Á̇ ̃ÂÌˡÏË ÒÓÓÚ‚ÂÚÒÚ‚Û ̨ ̆Ëı ÙÛÌ͈ËÈ ‚ ÚÓ ̃Í ‚ˉ‡1/q. œ‰ÒÚ‡‚ΡÂÚÒˇ ÂÒÚÂÒÚ‚ÂÌÌÓÈ Ú‡ÍÊ ‚ÓÁÏÓÊÌ